$( #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# Mathbox for Peter Mazsa #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# $) $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Contents =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) $( Notations (incl. symbol definitions) Preparatory theorems Tail Cartesian product (supplement) Cosets by ` R ` Relations Subset relations Reflexivity Converse reflexivity Symmetry Reflexivity and symmetry Transitivity Equivalence relations Domain quotients Equivalence relations on domain quotients Functions Disjoints vs. converse functions Partitions: disjoints on domain quotients Partition-Equivalence Theorems $) $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Notations (incl. symbol definitions) =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) $c ,~ $. $( Class of cosets symbol $) $c ~ $. $( Class of coelements symbol $) $c Rels $. $( The class of all relations $) $c _S $. $( The class of all subset relations $) $c Refs $. $( The class of all reflexive sets (used only once) $) $c RefRels $. $( The class of all reflexive relations $) $c RefRel $. $( Reflexive relation predicate $) $c CnvRefs $. $( The class of all converse reflexive sets (used only once) $) $c CnvRefRels $. $( The class of all converse reflexive relations $) $c CnvRefRel $. $( Converse reflexive relation predicate $) $c Syms $. $( The class of all symmetric sets (used only once) $) $c SymRels $. $( The class of all symmetric relations $) $c SymRel $. $( Symmetric relation predicate $) $c Trs $. $( The class of all transitive sets vs. the transitive class defined in ~ df-tr (used only once) $) $c TrRels $. $( The class of all transitive relations $) $c TrRel $. $( Transitive relation predicate $) $c EqvRels $. $( The class of all equivalence relations $) $c EqvRel $. $( Equivalence relation predicate $) $c ElEqvRel $. $( Elementhood equivalence relation predicate (used only for a placeholder definition) $) $c DomainQss $. $( The class of all domain quotients $) $c DomainQs $. $( Domain quotient predicate $) $c Ers $. $( The class of all equivalence relations on their domain quotients $) $c ErALTV $. $( Equivalence relation on its domain quotient predicate $) $c MembEr $. $( Membership equivalence relation predicate $) $c Funss $. $( The class of all function sets (used only once) $) $c FunsALTV $. $( The class of all functions, i.e., function relations $) $c FunALTV $. $( Function predicate $) $c Disjss $. $( The class of all disjoint sets (used only once) $) $c Disjs $. $( The class of all disjoints, i.e., disjoint relations $) $c Disj $. $( Disjoint predicate $) $c ElDisj $. $( Disjoint elementhood predicate $) $c Parts $. $( The class of all partitions, i.e., partition relations $) $c Part $. $( Partition predicate $) $c MembPart $. $( Membership partition predicate $) $( Extend the definition of a class to include the class of cosets by a class. (Read: the class of cosets by ` R ` .) $) ccoss $a class ,~ R $. $( Extend the definition of a class to include the class of coelements on a class. (Read: the class of coelements on ` A ` .) $) ccoels $a class ~ A $. $( Extend the definition of a class to include the relation class. $) crels $a class Rels $. $( Extend the definition of a class to include the subset class. $) cssr $a class _S $. $( Extend the definition of a class to include the reflexivity class. $) crefs $a class Refs $. $( Extend the definition of a class to include the reflexive relations class. $) crefrels $a class RefRels $. $( Extend the definition of a wff to include the reflexive relation predicate. (Read: ` R ` is a reflexive relation.) $) wrefrel $a wff RefRel R $. $( Extend the definition of a class to include the converse reflexivity class. $) ccnvrefs $a class CnvRefs $. $( Extend the definition of a class to include the converse reflexive relations class. $) ccnvrefrels $a class CnvRefRels $. $( Extend the definition of a wff to include the converse reflexive relation predicate. (Read: ` R ` is a converse reflexive relation.) $) wcnvrefrel $a wff CnvRefRel R $. $( Extend the definition of a class to include the symmetry class. $) csyms $a class Syms $. $( Extend the definition of a class to include the symmetry relations class. $) csymrels $a class SymRels $. $( Extend the definition of a wff to include the symmetry relation predicate. (Read: ` R ` is a symmetric relation.) $) wsymrel $a wff SymRel R $. $( Extend the definition of a class to include the transitivity class (but cf. the transitive class defined in ~ df-tr ). $) ctrs $a class Trs $. $( Extend the definition of a class to include the transitive relations class. $) ctrrels $a class TrRels $. $( Extend the definition of a wff to include the transitive relation predicate. (Read: ` R ` is a transitive relation.) $) wtrrel $a wff TrRel R $. $( Extend the definition of a class to include the equivalence relations class. $) ceqvrels $a class EqvRels $. $( Extend the definition of a wff to include the equivalence relation predicate. (Read: ` R ` is an equivalence relation.) $) weqvrel $a wff EqvRel R $. $( Extend the definition of a wff to include the elementhood equivalence relation predicate. (Read: the elementhood equivalence relation on ` A ` .) $) weleqvrel $a wff ElEqvRel A $. $( Extend the definition of a class to include the domain quotients class. $) cdmqss $a class DomainQss $. $( Extend the definition of a wff to include the domain quotient predicate. (Read: the domain quotient of ` R ` is ` A ` .) $) wdmqs $a wff R DomainQs A $. $( Extend the definition of a class to include the equivalence relations on domain quotients class. $) cers $a class Ers $. $( Extend the definition of a wff to include the equivalence relation on its domain quotient predicate. (Read: ` R ` is an equivalence relation on its domain quotient ` A ` .) $) werALTV $a wff R ErALTV A $. $( Extend the definition of a wff to include the membership equivalence relation predicate. (Read: the membership equivalence relation on ` A ` , or, the restricted elementhood equivalence relation on its domain quotient ` A ` .) $) wmember $a wff MembEr A $. $( Extend the definition of a class to include the function set class. $) cfunss $a class Funss $. $( Extend the definition of a class to include the functions class, i.e., the function relations class. $) cfunsALTV $a class FunsALTV $. $( Extend the definition of a wff to include the function predicate, i.e., the function relation predicate. (Read: ` F ` is a function.) $) wfunALTV $a wff FunALTV F $. $( Extend the definition of a class to include the disjoint set class. $) cdisjss $a class Disjss $. $( Extend the definition of a class to include the disjoints class, i.e., the disjoint relations class. $) cdisjs $a class Disjs $. $( Extend the definition of a wff to include the disjoint predicate, i.e., the disjoint relation predicate. (Read: ` R ` is a disjoint.) $) wdisjALTV $a wff Disj R $. $( Extend the definition of a wff to include the disjoint elementhood predicate, i.e., the disjoint elementhood relation predicate. (Read: the elements of ` A ` are disjoint.) $) weldisj $a wff ElDisj A $. $( Extend the definition of a class to include the partitions class, i.e., the partition relations class. $) cparts $a class Parts $. $( Extend the definition of a wff to include the partition predicate, i.e., the partition relation predicate. (Read: ` A ` is a partition by ` R ` .) $) wpart $a wff R Part A $. $( Extend the definition of a wff to include the membership partition predicate, i.e., the membership partition relation predicate. (Read: ` A ` is a membership partition.) $) wmembpart $a wff MembPart A $. $( /******* Symbol definitions *******/ htmldef ",~" as " ,~ "; althtmldef ",~" as ' ≀ '; latexdef ",~" as "\wr"; htmldef "~" as " ~ "; althtmldef "~" as ' ∼ '; latexdef "~" as "\sim"; htmldef "Rels" as ' Rels '; althtmldef "Rels" as ' Rels '; latexdef "Rels" as "{\rm Rels}"; htmldef "_S" as " _S "; althtmldef "_S" as ' S '; latexdef "_S" as "{\rm S}"; htmldef "Refs" as ' Refs '; althtmldef "Refs" as ' Refs '; latexdef "Refs" as "{\rm Refs}"; htmldef "RefRels" as ' RefRels '; althtmldef "RefRels" as ' RefRels '; latexdef "RefRels" as "{\rm RefRels}"; htmldef "RefRel" as ' RefRel '; althtmldef "RefRel" as ' RefRel '; latexdef "RefRel" as "{\rm RefRel}"; htmldef "CnvRefs" as ' CnvRefs '; althtmldef "CnvRefs" as ' CnvRefs '; latexdef "CnvRefs" as "{\rm CnvRefs}"; htmldef "CnvRefRels" as ' CnvRefRels '; althtmldef "CnvRefRels" as ' CnvRefRels '; latexdef "CnvRefRels" as "{\rm CnvRefRels}"; htmldef "CnvRefRel" as ' CnvRefRel '; althtmldef "CnvRefRel" as ' CnvRefRel '; latexdef "CnvRefRel" as "{\rm CnvRefRel}"; htmldef "Syms" as ' Syms '; althtmldef "Syms" as ' Syms '; latexdef "Syms" as "{\rm Syms}"; htmldef "SymRels" as ' SymRels '; althtmldef "SymRels" as ' SymRels '; latexdef "SymRels" as "{\rm SymRels}"; htmldef "SymRel" as ' SymRel '; althtmldef "SymRel" as ' SymRel '; latexdef "SymRel" as "{\rm SymRel}"; htmldef "Trs" as ' Trs '; althtmldef "Trs" as ' Trs '; latexdef "Trs" as "{\rm Trs}"; htmldef "TrRels" as ' TrRels '; althtmldef "TrRels" as ' TrRels '; latexdef "TrRels" as "{\rm TrRels}"; htmldef "TrRel" as ' TrRel '; althtmldef "TrRel" as ' TrRel '; latexdef "TrRel" as "{\rm TrRel}"; htmldef "EqvRels" as ' EqvRels '; althtmldef "EqvRels" as ' EqvRels '; latexdef "EqvRels" as "{\rm EqvRels}"; htmldef "EqvRel" as ' EqvRel '; althtmldef "EqvRel" as ' EqvRel '; latexdef "EqvRel" as "{\rm EqvRel}"; htmldef "ElEqvRel" as ' ElEqvRel '; althtmldef "ElEqvRel" as ' ElEqvRel '; latexdef "ElEqvRel" as "{\rm ElEqvRel}"; htmldef "DomainQss" as ' DomainQss '; althtmldef "DomainQss" as ' DomainQss '; latexdef "DomainQss" as "{\rm DomainQss}"; htmldef "DomainQs" as ' DomainQs '; althtmldef "DomainQs" as ' DomainQs '; latexdef "DomainQs" as "{\rm DomainQs}"; htmldef "Ers" as ' Ers '; althtmldef "Ers" as ' Ers '; latexdef "Ers" as "{\rm Ers}"; htmldef "ErALTV" as ' ErALTV '; althtmldef "ErALTV" as ' ErALTV '; latexdef "ErALTV" as "{\rm ErALTV}"; htmldef "MembEr" as ' MembEr '; althtmldef "MembEr" as ' MembEr '; latexdef "MembEr" as "{\rm MembEr}"; htmldef "Funss" as ' Funss '; althtmldef "Funss" as ' Funss '; latexdef "Funss" as "{\rm Funss}"; htmldef "FunsALTV" as ' FunsALTV '; althtmldef "FunsALTV" as ' FunsALTV '; latexdef "FunsALTV" as "{\rm FunsALTV}"; htmldef "FunALTV" as ' FunALTV '; althtmldef "FunALTV" as ' FunALTV '; latexdef "FunALTV" as "{\rm FunALTV}"; htmldef "Disjss" as ' Disjss '; althtmldef "Disjss" as ' Disjss '; latexdef "Disjss" as "{\rm Disjss}"; htmldef "Disjs" as ' Disjs '; althtmldef "Disjs" as ' Disjs '; latexdef "Disjs" as "{\rm Disjs}"; htmldef "Disj" as ' Disj '; althtmldef "Disj" as ' Disj '; latexdef "Disj" as "{\rm Disj}"; htmldef "ElDisj" as ' ElDisj '; althtmldef "ElDisj" as ' ElDisj '; latexdef "ElDisj" as "{\rm ElDisj}"; htmldef "Parts" as ' Parts '; althtmldef "Parts" as ' Parts '; latexdef "Parts" as "{\rm Parts}"; htmldef "Part" as ' Part '; althtmldef "Part" as ' Part '; latexdef "Part" as "{\rm Part}"; htmldef "MembPart" as ' MembPart '; althtmldef "MembPart" as ' MembPart '; latexdef "MembPart" as "{\rm MembPart}"; $) $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Preparatory theorems =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) ${ elv.1 $e |- ( x e. _V -> ph ) $. $( New way ( ~ elv , ~ el2v theorems and ~ el3v theorems) to shorten some proofs. Inference forms (with ` $e |- A e. _V $. ` ) of the general theorems (with ` $p |- ( A e. V -> ` ) may be superfluous. (Contributed by Peter Mazsa, 13-Oct-2018.) $) elv $p |- ph $= ( cv cvv wcel vex ax-mp ) BDEFABGCH $. $} ${ el2v.1 $e |- ( ( x e. _V /\ y e. _V ) -> ph ) $. $( New way ( ~ elv , ~ el2v theorems and ~ el3v theorems) to shorten some proofs. Inference forms (with ` $e |- A e. _V $. ` and ` $e |- B e. _V $. ` ) of the general theorems (with ` $p |- ( ( A e. V /\ B e. W ) -> ` ) may be superfluous. (Contributed by Peter Mazsa, 13-Oct-2018.) $) el2v $p |- ph $= ( cv cvv wcel vex mp2an ) BEFGCEFGABHCHDI $. $} ${ el2v1.1 $e |- ( ( x e. _V /\ ph ) -> ps ) $. $( New way ( ~ elv , ~ el2v theorems and ~ el3v theorems) to shorten some proofs. (Contributed by Peter Mazsa, 23-Oct-2018.) $) el2v1 $p |- ( ph -> ps ) $= ( cv cvv wcel vex mpan ) CEFGABCHDI $. $} ${ el2v2.1 $e |- ( ( ph /\ y e. _V ) -> ps ) $. $( New way ( ~ elv , ~ el2v theorems and ~ el3v theorems) to shorten some proofs. (Contributed by Peter Mazsa, 23-Oct-2018.) $) el2v2 $p |- ( ph -> ps ) $= ( cv cvv wcel vex mpan2 ) ACEFGBCHDI $. $} ${ el3v.1 $e |- ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ph ) $. $( New way ( ~ elv , ~ el2v theorems and ~ el3v theorems) to shorten some proofs. Inference forms (with ` $e |- A e. _V $. ` , ` $e |- B e. _V $. ` and ` $e |- C e. _V $. ` ) of the general theorems (with ` $p |- ( ( A e. V /\ B e. W /\ C e. X ) -> ` ) may be superfluous. (Contributed by Peter Mazsa, 13-Oct-2018.) $) el3v $p |- ph $= ( cv cvv wcel vex mp3an ) BFGHCFGHDFGHABICIDIEJ $. $} ${ el3v1.1 $e |- ( ( x e. _V /\ ps /\ ch ) -> th ) $. $( New way ( ~ elv , ~ el2v theorems and ~ el3v theorems) to shorten some proofs. (Contributed by Peter Mazsa, 16-Oct-2020.) $) el3v1 $p |- ( ( ps /\ ch ) -> th ) $= ( cv cvv wcel vex mp3an1 ) DFGHABCDIEJ $. $} ${ el3v2.1 $e |- ( ( ph /\ y e. _V /\ ch ) -> th ) $. $( New way ( ~ elv , ~ el2v theorems and ~ el3v theorems) to shorten some proofs. (Contributed by Peter Mazsa, 16-Oct-2020.) $) el3v2 $p |- ( ( ph /\ ch ) -> th ) $= ( cv cvv wcel vex mp3an2 ) ADFGHBCDIEJ $. $} ${ el3v3.1 $e |- ( ( ph /\ ps /\ z e. _V ) -> th ) $. $( New way ( ~ elv , ~ el2v theorems and ~ el3v theorems) to shorten some proofs. (Contributed by Peter Mazsa, 16-Oct-2020.) $) el3v3 $p |- ( ( ph /\ ps ) -> th ) $= ( cv cvv wcel vex mp3an3 ) ABDFGHCDIEJ $. $} ${ el3v12.1 $e |- ( ( x e. _V /\ y e. _V /\ ch ) -> th ) $. $( New way ( ~ elv , ~ el2v theorems and ~ el3v theorems) to shorten some proofs. (Contributed by Peter Mazsa, 11-Jul-2021.) $) el3v12 $p |- ( ch -> th ) $= ( cv cvv wcel el3v1 el2v1 ) ABDDFGHABCEIJ $. $} ${ el3v13.1 $e |- ( ( x e. _V /\ ps /\ z e. _V ) -> th ) $. $( New way ( ~ elv , ~ el2v theorems and ~ el3v theorems) to shorten some proofs. (Contributed by Peter Mazsa, 11-Jul-2021.) $) el3v13 $p |- ( ps -> th ) $= ( cv cvv wcel el3v3 el2v1 ) ABCCFGHABDEIJ $. $} ${ el3v23.1 $e |- ( ( ph /\ y e. _V /\ z e. _V ) -> th ) $. $( New way ( ~ elv , ~ el2v theorems and ~ el3v theorems) to shorten some proofs. (Contributed by Peter Mazsa, 11-Jul-2021.) $) el3v23 $p |- ( ph -> th ) $= ( cv cvv wcel el3v3 el2v2 ) ABCACFGHBDEIJ $. $} ${ bian.1 $e |- ( ph <-> ( ch /\ ps ) ) $. $( Commuting conjunction in a biconditional. (Contributed by Peter Mazsa, 17-Jun-2018.) $) bian $p |- ( ph <-> ( ps /\ ch ) ) $= ( wa ancom bitr4i ) ACBEBCEDBCFG $. $} ${ biand.1 $e |- ( ph -> ( ps <-> ( th /\ ch ) ) ) $. $( Commuting conjunction in a biconditional, deduction form. (Contributed by Peter Mazsa, 3-Oct-2018.) $) biand $p |- ( ph -> ( ps <-> ( ch /\ th ) ) ) $= ( wa wb ancom bibi2i sylib ) ABDCFZGBCDFZGEKLBDCHIJ $. $} ${ anbi2ri.1 $e |- ( ph <-> ps ) $. $( Introduce a left and the same right conjunct to the sides of a logical equivalence. (Contributed by Peter Mazsa, 7-Mar-2020.) $) anbi2ri $p |- ( ( ch /\ ph ) <-> ( ps /\ ch ) ) $= ( wa anbi2i bian ) CAEBCABCDFG $. $} ${ anbi2rid.1 $e |- ( ph -> ( ps <-> ch ) ) $. $( Introduce a left and the same right conjunct to the sides of a logical equivalence, deduction form. (Contributed by Peter Mazsa, 22-May-2021.) $) anbi2rid $p |- ( ph -> ( ( th /\ ps ) <-> ( ch /\ th ) ) ) $= ( wb wa anbi2 biand syl ) ABCFZDBGZCDGFEKLCDBCDHIJ $. $} $( Double commutation in conjunction. (Contributed by Peter Mazsa, 27-Jun-2019.) $) an2anr $p |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> ( ( ps /\ ph ) /\ ( th /\ ch ) ) ) $= ( wa ancom anbi12i ) ABEBAECDEDCEABFCDFG $. $( Multiple commutations in conjunction. (Contributed by Peter Mazsa, 7-Mar-2020.) $) anan $p |- ( ( ( ( ph /\ ps ) /\ ch ) /\ ( ( ph /\ th ) /\ ta ) ) <-> ( ( ps /\ th ) /\ ( ph /\ ( ch /\ ta ) ) ) ) $= ( wa an4 anandi ancom bitr3i anbi1i anass 3bitri ) ABFZCFADFZEFFNOFZCEFZFBD FZAFZQFRAQFFNCOEGPSQPARFSABDHARIJKRAQLM $. ${ triantru3.1 $e |- ph $. triantru3.2 $e |- ps $. $( A wff is equivalent to its conjunctions with truths. (Contributed by Peter Mazsa, 30-Nov-2018.) $) triantru3 $p |- ( ch <-> ( ph /\ ps /\ ch ) ) $= ( wa w3a biantrur 3anass 3bitr4i ) BCFZAKFCABCGAKDHBCEHABCIJ $. $} $( Substitution of equal classes into elementhood relation. (Contributed by Peter Mazsa, 22-Jul-2017.) $) eqeltr $p |- ( ( A = B /\ B e. C ) -> A e. C ) $= ( wceq wcel eleq1 biimpar ) ABDACEBCEABCFG $. $( Substitution of equal classes into elementhood relation. (Contributed by Peter Mazsa, 17-Jul-2019.) $) eqelb $p |- ( ( A = B /\ A e. C ) <-> ( A = B /\ B e. C ) ) $= ( wceq wcel wa simpl eqeltr jca eqcom anbi1i 3imtr3i impbii ) ABDZACEZFZNBC EZFZBADZOFZSQFPRTSQSOGBACHISNOBAJZKSNQUAKLRNONQGABCHIM $. ${ eqeqan1d.1 $e |- ( ph -> A = B ) $. $( Implication of introducing a new equality. (Contributed by Peter Mazsa, 17-Apr-2019.) $) eqeqan1d $p |- ( ( ph /\ C = D ) -> ( A = C <-> B = D ) ) $= ( wceq wb eqeq12 sylan ) ABCGDEGBDGCEGHFBCDEIJ $. $} ${ eqeqan2d.1 $e |- ( ph -> C = D ) $. $( Implication of introducing a new equality. (Contributed by Peter Mazsa, 17-Apr-2019.) $) eqeqan2d $p |- ( ( A = B /\ ph ) -> ( A = C <-> B = D ) ) $= ( wceq wb eqeq12 sylan2 ) ABCGDEGBDGCEGHFBCDEIJ $. $} $( Two ways of saying that two classes are disjoint (when ` C = (/) ` : ` ( ( A i^i B ) = (/) <-> ( B i^i A ) = (/) ) ` ). (Contributed by Peter Mazsa, 22-Mar-2017.) $) ineq $p |- ( ( A i^i B ) = C <-> ( B i^i A ) = C ) $= ( cin incom eqeq1i ) ABDBADCABEF $. ${ ineqi.1 $e |- ( A i^i B ) = C $. $( Disjointness inference (when ` C = (/) ` ), inference form of ~ ineq . (Contributed by Peter Mazsa, 26-Mar-2017.) $) ineqi $p |- ( B i^i A ) = C $= ( cin wceq ineq mpbi ) ABECFBAECFDABCGH $. $} $( Two ways of expressing the restriction of an intersection. (Contributed by Peter Mazsa, 5-Jun-2021.) $) inres2 $p |- ( ( R |` A ) i^i S ) = ( ( R i^i S ) |` A ) $= ( cres cin inres ineqi incom reseq1i eqtr4i ) BADZCECBEZADZBCEZADCKMCBAFGNL ABCHIJ $. $( Subclass theorem for binary relation. (Contributed by Peter Mazsa, 11-Nov-2019.) $) brss $p |- ( R C_ S -> ( A R B -> A S B ) ) $= ( wss cop wcel wbr ssel df-br 3imtr4g ) CDEABFZCGLDGABCHABDHCDLIABCJABDJK $. ${ brssi.1 $e |- R C_ S $. $( Subclass theorem for binary relation, inference version. (Contributed by Peter Mazsa, 11-Nov-2019.) $) brssi $p |- ( A R B -> A S B ) $= ( wss wbr wi brss ax-mp ) CDFABCGABDGHEABCDIJ $. $} ${ brssd.1 $e |- ( ph -> R C_ S ) $. $( Subclass theorem for binary relation, deduction version. (Contributed by Peter Mazsa, 11-Nov-2019.) $) brssd $p |- ( ph -> ( A R B -> A S B ) ) $= ( wss wbr wi brss syl ) ADEGBCDHBCEHIFBCDEJK $. $} $( Equality theorem for composition of two classes. (Contributed by Peter Mazsa, 23-Sep-2021.) $) coideq $p |- ( A = B -> ( A o. A ) = ( B o. B ) ) $= ( wceq ccom coeq1 coeq2 eqtrd ) ABCAADBADBBDABAEABBFG $. $( All elements of a class are elements of the class (predates ~ ralel ). (Contributed by Peter Mazsa, 22-Dec-2018.) $) ralid $p |- A. x e. A x e. A $= ( cv wcel wral wi wal id ax-gen df-ral mpbir ) ACBDZABELLFZAGMALHILABJK $. $( Cancellation law for restriction. (Contributed by Peter Mazsa, 30-Dec-2018.) $) ralanid $p |- ( A. x e. A ( x e. A /\ ph ) <-> A. x e. A ph ) $= ( cv wcel wi wal wa wral anclb albii df-ral 3bitr4ri ) BDCEZAFZBGNNAHZFZBGA BCIPBCIOQBNAJKABCLPBCLM $. $( If there is no case where wff is true, it is true for at most one case. (Contributed by Peter Mazsa, 27-Sep-2021.) $) nexmo $p |- ( -. E. x ph -> E* x ph ) $= ( wex wn weu wi wmo pm2.21 df-mo sylibr ) ABCZDKABEZFABGKLHABIJ $. ${ 3albii.1 $e |- ( ph <-> ps ) $. $( Inference adding three universal quantifiers to both sides of an equivalence. (Contributed by Peter Mazsa, 10-Aug-2018.) $) 3albii $p |- ( A. x A. y A. z ph <-> A. x A. y A. z ps ) $= ( wal 2albii albii ) AEGDGBEGDGCABDEFHI $. $} ${ 3ralbii.1 $e |- ( ph <-> ps ) $. $( Inference adding three restricted universal quantifiers to both sides of an equivalence. (Contributed by Peter Mazsa, 25-Jul-2019.) $) 3ralbii $p |- ( A. x e. A A. y e. B A. z e. C ph <-> A. x e. A A. y e. B A. z e. C ps ) $= ( wral 2ralbii ralbii ) AEHJDGJBEHJDGJCFABDEGHIKL $. $} ${ rabbiia2.1 $e |- ( ph <-> ps ) $. $( Equivalent wff's correspond to equal restricted class abstractions. (Contributed by Peter Mazsa, 1-Nov-2019.) $) rabbiia2 $p |- { x e. A | ph } = { x e. A | ps } $= ( wb cv wcel a1i rabbiia ) ABCDABFCGDHEIJ $. $} ${ rabbieq.1 $e |- B = { x e. A | ph } $. rabbieq.2 $e |- ( ph <-> ps ) $. $( Equivalent wff's correspond to restricted class abstractions which are equal with the same class. (Contributed by Peter Mazsa, 8-Jul-2019.) $) rabbieq $p |- B = { x e. A | ps } $= ( crab rabbiia2 eqtri ) EACDHBCDHFABCDGIJ $. $} ${ rabimbieq.1 $e |- B = { x e. A | ph } $. rabimbieq.2 $e |- ( x e. A -> ( ph <-> ps ) ) $. $( Restricted equivalent wff's correspond to restricted class abstractions which are equal with the same class. (Contributed by Peter Mazsa, 22-Jul-2021.) $) rabimbieq $p |- B = { x e. A | ps } $= ( crab rabbiia eqtri ) EACDHBCDHFABCDGIJ $. $} ${ $d C x $. abeqin.1 $e |- A = ( B i^i C ) $. abeqin.2 $e |- B = { x | ph } $. $( Intersection with class abstraction. (Contributed by Peter Mazsa, 21-Jul-2021.) $) abeqin $p |- A = { x e. C | ph } $= ( cin cab crab ineq1i dfrab2 3eqtr4i ) DEHABIZEHCABEJDNEGKFABELM $. $} ${ $d C x $. abeqinbi.1 $e |- A = ( B i^i C ) $. abeqinbi.2 $e |- B = { x | ph } $. abeqinbi.3 $e |- ( x e. C -> ( ph <-> ps ) ) $. $( Intersection with class abstraction and equivalent wff's. (Contributed by Peter Mazsa, 21-Jul-2021.) $) abeqinbi $p |- A = { x e. C | ps } $= ( crab abeqin rabbiia eqtri ) DACFJBCFJACDEFGHKABCFILM $. $} ${ $d A x $. $d C x $. $d ps x $. rabeqel.1 $e |- B = { x e. A | ph } $. rabeqel.2 $e |- ( x = C -> ( ph <-> ps ) ) $. $( Class element of a restricted class abstraction. (Contributed by Peter Mazsa, 24-Jul-2021.) $) rabeqel $p |- ( C e. B <-> ( ps /\ C e. A ) ) $= ( wcel crab wa eleq2i elrab bitri bian ) FEIZBFDIZPFACDJZIQBKERFGLABCFDHM NO $. $} ${ $d A u v $. $d B u v $. $d u v x y $. eqrelf.1 $e |- F/_ x A $. eqrelf.2 $e |- F/_ x B $. eqrelf.3 $e |- F/_ y A $. eqrelf.4 $e |- F/_ y B $. $( The equality connective between relations. (Contributed by Peter Mazsa, 25-Jun-2019.) $) eqrelf $p |- ( ( Rel A /\ Rel B ) -> ( A = B <-> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. B ) ) ) $= ( vu vv wrel wa wceq cv cop wcel wb wal nfcv nfel eqrel nfv opeq12 eleq1d nfop nfbi bibi12d cbval2 syl6bbr ) CKDKLCDMINZJNZOZCPZULDPZQZJRIRANZBNZOZ CPZURDPZQZBRARIJCDUAVAUOABIJVAIUBVAJUBUMUNAAULCAUJUKAUJSAUKSUEZETAULDVBFT UFUMUNBBULCBUJUKBUJSBUKSUEZGTBULDVCHTUFUPUJMUQUKMLZUSUMUTUNVDURULCUPUQUJU KUCZUDVDURULDVEUDUGUHUI $. $} $( Elementhood in a converse ` R ` -coset when ` R ` is a relation. (Contributed by Peter Mazsa, 9-Dec-2018.) $) releleccnv $p |- ( Rel R -> ( A e. [ B ] `' R <-> A R B ) ) $= ( ccnv cec wcel wbr wrel wb relcnv relelec ax-mp relbrcnvg syl5bb ) ABCDZEF ZBAOGZCHABCGOHPQICJABOKLBACMN $. ${ $d A x $. $d B x $. $d R x $. $d S x $. $( Equality of converse ` R ` -coset and converse ` S ` -coset when ` R ` and ` S ` are relations. (Contributed by Peter Mazsa, 27-Jul-2019.) $) releccnveq $p |- ( ( Rel R /\ Rel S ) -> ( [ A ] `' R = [ B ] `' S <-> A. x ( x R A <-> x S B ) ) ) $= ( ccnv cec wceq cv wcel wb wal wrel wbr dfcleq releleccnv bi2bian9 albidv wa syl5bb ) BDFGZCEFGZHAIZUAJZUCUBJZKZALDMZEMZSZUCBDNZUCCENZKZALAUAUBOUIU FULAUGUDUJUHUEUKUCBDPUCCEPQRT $. $} $( Negated elementhood of ordered pair. (Contributed by Peter Mazsa, 14-Jan-2019.) $) opelvvdif $p |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. e. ( ( _V X. _V ) \ R ) <-> -. <. A , B >. e. R ) ) $= ( wcel wa cop wn cvv cxp cdif opelvvg biantrurd eldif syl6rbbr ) ADFBEFGZAB HZCFIZRJJKZFZSGRTCLFQUASABDEMNRTCOP $. ${ $d x y $. $( Ordered-pair class abstraction defined by a negation. (Contributed by Peter Mazsa, 25-Jun-2019.) $) vvdifopab $p |- ( ( _V X. _V ) \ { <. x , y >. | ph } ) = { <. x , y >. | -. ph } $= ( cvv cxp copab cdif wn wceq cv cop wcel wb wal opabid wrel nfopab1 nfdif nfcv nfopab2 notbii opelvvdif el2v 3bitr4i gen2 relxp reldif ax-mp eqrelf relopab mp2an mpbir ) DDEZABCFZGZAHZBCFZIZBJZCJZKZUOLZVAUQLZMZCNBNZVDBCVA UNLZHZUPVBVCVFAABCOUAVBVGMBCUSUTUNDDUBUCUPBCOUDUEUOPZUQPURVEMUMPVHDDUFUMU NUGUHUPBCUJBCUOUQBUMUNBUMSABCQRUPBCQCUMUNCUMSABCTRUPBCTUIUKUL $. $} $( Binary relation with universal complement is the negation of the relation. (Contributed by Peter Mazsa, 1-Jul-2018.) $) brvdif $p |- ( A ( _V \ R ) B <-> -. A R B ) $= ( cvv cdif wbr wn wa brdif brv biantrur bitr4i ) ABDCEFABDFZABCFGZHNABDCIMN ABJKL $. $( Binary relation with universal complement. (Contributed by Peter Mazsa, 14-Jul-2018.) $) brvdif2 $p |- ( A ( _V \ R ) B <-> -. <. A , B >. e. R ) $= ( cvv cdif wbr wn cop wcel brvdif df-br notbii bitri ) ABDCEFABCFZGABHCIZGA BCJNOABCKLM $. $( Binary relation with the complement under the universal class of ordered pairs. (Contributed by Peter Mazsa, 9-Nov-2018.) $) brvvdif $p |- ( ( A e. V /\ B e. W ) -> ( A ( ( _V X. _V ) \ R ) B <-> -. A R B ) ) $= ( wcel wa cop cvv cxp cdif wn wbr opelvvdif df-br notbii 3bitr4g ) ADFBEFGA BHZIIJCKZFRCFZLABSMABCMZLABCDENABSOUATABCOPQ $. $( Binary relation with the complement under the universal class of ordered pairs is the same as with universal complement. (Contributed by Peter Mazsa, 28-Nov-2018.) $) brvbrvvdif $p |- ( ( A e. V /\ B e. W ) -> ( A ( ( _V X. _V ) \ R ) B <-> A ( _V \ R ) B ) ) $= ( wcel wa cvv cxp cdif wbr wn brvvdif brvdif syl6bbr ) ADFBEFGABHHICJKABCKL ABHCJKABCDEMABCNO $. $( The converse of the binary epsilon relation. (Contributed by Peter Mazsa, 30-Jan-2018.) $) brcnvep $p |- ( A e. V -> ( A `' _E B <-> B e. A ) ) $= ( cep ccnv wbr wcel rele relbrcnv epelg syl5bb ) ABDEFBADFACGBAGABDHIBACJK $. $( Elementhood in the ` R ` -coset of ` A ` . Theorem 72 of [Suppes] p. 82. (I think we should replace ~ elecg with this original form of Suppes. Peter Mazsa) (Contributed by Mario Carneiro, 9-Jul-2014.) $) elecALTV $p |- ( ( A e. V /\ B e. W ) -> ( B e. [ A ] R <-> A R B ) ) $= ( wcel wa csn cima cop cec wbr elimasng df-ec eleq2i df-br 3bitr4g ) ADFBEF GBCAHIZFABJCFBACKZFABCLCABDEMSRBACNOABCPQ $. $( Ordered pair elementhood in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.) $) opelresALTV $p |- ( C e. V -> ( <. B , C >. e. ( R |` A ) <-> ( B e. A /\ <. B , C >. e. R ) ) ) $= ( cop cres wcel cvv cxp cin df-res eleq2i elin bitri biantrurd biand opelxp wa elex syl6rbbr anbi2rid syl5bb ) BCFZDAGZHZUDDHZUDAIJZHZSZCEHZBAHZUGSUFUD DUHKZHUJUEUMUDDALMUDDUHNOUKUIULUGUKULULCIHZSUIUKULULUNUKUNULCETPQBCAIRUAUBU C $. $( Binary relation on a restriction. (Contributed by Mario Carneiro, 4-Nov-2015.) $) brresALTV $p |- ( C e. V -> ( B ( R |` A ) C <-> ( B e. A /\ B R C ) ) ) $= ( wcel cop cres wa wbr opelresALTV df-br anbi2i 3bitr4g ) CEFBCGZDAHZFBAFZO DFZIBCPJQBCDJZIABCDEKBCPLSRQBCDLMN $. ${ $d A x y $. $d B x y $. $d C x y $. $( Restricted converse epsilon binary relation. (Contributed by Peter Mazsa, 10-Feb-2018.) $) brcnvepres $p |- ( ( B e. V /\ C e. W ) -> ( B ( `' _E |` A ) C <-> ( B e. A /\ C e. B ) ) ) $= ( wcel cep ccnv cres wbr wa brresALTV brcnvep anbi2d sylan9bbr ) CEFBCGHZ AIJBAFZBCPJZKBDFZQCBFZKABCPELSRTQBCDMNO $. $} $( Intersection with cross product binary relation . (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.) $) brinxp2ALTV $p |- ( C ( R i^i ( A X. B ) ) D <-> ( ( C e. A /\ D e. B ) /\ C R D ) ) $= ( cxp cin wbr wa wcel brin ancom brxp anbi1i 3bitri ) CDEABFZGHCDEHZCDPHZIR QICAJDBJIZQICDEPKQRLRSQCDABMNO $. $( Binary relation on a restriction. (Contributed by Peter Mazsa, 2-Jan-2019.) (Revised by Peter Mazsa, 16-Dec-2021.) $) brres2 $p |- ( B ( R |` A ) C <-> B ( R i^i ( A X. ran ( R |` A ) ) ) C ) $= ( cres crn wcel wbr wa cxp cin brresALTV pm5.32i wrel relelrn mpan pm4.71ri relres w3a brinxp2ALTV df-3an 3anan12 3bitr2i 3bitr4i ) CDAEZFZGZBCUEHZIUGB AGZBCDHZIZIZUHBCDAUFJKHZUGUHUKABCDUFLMUHUGUENUHUGDARBCUEOPQUMUIUGIUJIUIUGUJ SULAUFBCDTUIUGUJUAUIUGUJUBUCUD $. ${ $d A y $. $d B y $. $d R y $. $( Elementhood in the domain of a restriction. (Contributed by Peter Mazsa, 9-Jan-2019.) $) eldmres $p |- ( B e. V -> ( B e. dom ( R |` A ) <-> ( B e. A /\ E. y B R y ) ) ) $= ( wcel cres cdm cv wbr wex wa eldmg cvv brresALTV elv exbii 19.42v bitri wb syl6bb ) CEFCDBGZHFCAIZUBJZAKZCBFZCUCDJZAKLZACUBEMUEUFUGLZAKUHUDUIAUDU ITABCUCDNOPQUFUGARSUA $. $} ${ $d A y $. $d R y $. $d V y $. $( Elementhood in a domain. (Contributed by Peter Mazsa, 24-Oct-2018.) $) eldm4 $p |- ( A e. V -> ( A e. dom R <-> E. y y e. [ A ] R ) ) $= ( wcel cdm cv wbr wex cec eldmg wb cvv elecALTV el2v2 exbidv bitr4d ) BDE ZBCFEBAGZCHZAISBCJEZAIABCDKRUATARUATLABSCDMNOPQ $. $} ${ $d A y $. $d B y $. $d R y $. $d V y $. $( Elementhood in the domain of a restriction. (Contributed by Peter Mazsa, 21-Aug-2020.) $) eldmres2 $p |- ( B e. V -> ( B e. dom ( R |` A ) <-> ( B e. A /\ E. y y e. [ B ] R ) ) ) $= ( wcel cres cdm cv wbr wex wa cec eldmres eldmg eldm4 bitr3d anbi2d bitrd ) CEFZCDBGHFCBFZCAIZDJAKZLUAUBCDMFAKZLABCDENTUCUDUATCDHFUCUDACDEOACDEPQRS $. $} ${ eceq1i.1 $e |- A = B $. $( Equality theorem for ` C ` -coset of ` A ` and ` C ` -coset of ` B ` , inference version. (Contributed by Peter Mazsa, 11-May-2021.) $) eceq1i $p |- [ A ] C = [ B ] C $= ( wceq cec eceq1 ax-mp ) ABEACFBCFEDABCGH $. $} ${ eceq2i.1 $e |- A = B $. $( Equality theorem for the ` A ` -coset and ` B ` -coset of ` C ` , inference version. (Contributed by Peter Mazsa, 11-May-2021.) $) eceq2i $p |- [ C ] A = [ C ] B $= ( wceq cec eceq2 ax-mp ) ABECAFCBFEDABCGH $. $} ${ eceq2d.1 $e |- ( ph -> A = B ) $. $( Equality theorem for the ` A ` -coset and ` B ` -coset of ` C ` , deduction version. (Contributed by Peter Mazsa, 23-Apr-2021.) $) eceq2d $p |- ( ph -> [ C ] A = [ C ] B ) $= ( wceq cec eceq2 syl ) ABCFDBGDCGFEBCDHI $. $} $( Elementhood in the restricted coset of ` B ` . (Contributed by Peter Mazsa, 21-Sep-2018.) $) elecres $p |- ( C e. V -> ( C e. [ B ] ( R |` A ) <-> ( B e. A /\ B R C ) ) ) $= ( cres cec wcel wbr wa wrel wb relres relelec ax-mp brresALTV syl5bb ) CBDA FZGHZBCRIZCEHBAHBCDIJRKSTLDAMCBRNOABCDEPQ $. ${ $d A x $. $d B x $. $d R x $. $( Restricted coset of ` B ` . (Contributed by Peter Mazsa, 9-Dec-2018.) $) ecres $p |- [ B ] ( R |` A ) = { x | ( B e. A /\ B R x ) } $= ( wcel cv wbr wa cres cec wb cvv elecres elv abbi2i ) CBECAFZDGHZACDBIJZP REQKABCPDLMNO $. $} ${ $d A y $. $d B y $. $d R y $. $( The restricted coset of ` B ` when ` B ` is an element of the restriction. (Contributed by Peter Mazsa, 16-Oct-2018.) $) ecres2 $p |- ( B e. A -> [ B ] ( R |` A ) = [ B ] R ) $= ( vy wcel cres cec cv wbr cab wa wb cvv elecres baib abbi2dv dfec2 eqtr4d elv ) BAEZBCAFGZBDHZCIZDJBCGTUCDUAUBUAEZTUCUDTUCKLDABUBCMNSOPDBCAQR $. $} ${ $d A x $. $d B x $. $d V x $. $( Restricted converse epsilon coset of ` B ` . (Contributed by Peter Mazsa, 11-Feb-2018.) (Revised by Peter Mazsa, 21-Oct-2021.) $) eccnvepres $p |- ( B e. V -> [ B ] ( `' _E |` A ) = { x e. B | B e. A } ) $= ( wcel cv cep ccnv wbr wa cab cres cec crab brcnvep anbi2rid abbidv ecres df-rab 3eqtr4g ) CDEZCBEZCAFZGHZIZJZAKUCCEZUBJZAKCUDBLMUBACNUAUFUHAUAUEUG UBCUCDOPQABCUDRUBACST $. $} $( Elementhood in the converse epsilon coset of ` A ` is elementhood in ` A ` . (Contributed by Peter Mazsa, 27-Jan-2019.) $) eleccnvep $p |- ( A e. V -> ( B e. [ A ] `' _E <-> B e. A ) ) $= ( cep ccnv cec wcel wbr wrel wb relcnv relelec ax-mp brcnvep syl5bb ) BADEZ FGZABPHZACGBAGPIQRJDKBAPLMABCNO $. ${ $d A x $. $d V x $. $( The converse epsilon coset of a set is the set. (Contributed by Peter Mazsa, 27-Jan-2019.) $) eccnvep $p |- ( A e. V -> [ A ] `' _E = A ) $= ( vx wcel cep ccnv cec cv cab eleccnvep abbi2dv abid2 syl6eq ) ABDZAEFGZC HZADZCIANQCOAPBJKCALM $. $} $( Property of epsilon relation, cf. ~ extid , ~ extssr and the comment of ~ df-ssr . (Contributed by Peter Mazsa, 10-Jul-2019.) $) extep $p |- ( ( A e. V /\ B e. W ) -> ( [ A ] `' _E = [ B ] `' _E <-> A = B ) ) $= ( wcel cep ccnv cec eccnvep eqeqan12d ) ACEBDEAFGZHABKHBACIBDIJ $. $( The restricted converse epsilon coset of an element of the restriction is the element itself. (Contributed by Peter Mazsa, 16-Jul-2019.) $) eccnvepres2 $p |- ( B e. A -> [ B ] ( `' _E |` A ) = B ) $= ( wcel cep ccnv cres cec wceq ecres2 eccnvep eqeq2d mpbid ) BACZBDEZAFGZBNG ZHOBHABNIMPBOBAJKL $. $( Condition for a restricted converse epsilon coset of a set to be the set itself. (Contributed by Peter Mazsa, 11-May-2021.) $) eccnvepres3 $p |- ( B e. dom ( `' _E |` A ) -> [ B ] ( `' _E |` A ) = B ) $= ( cep ccnv cres cdm wcel cec resdmres eceq2i eccnvepres2 syl5eqr ) BCDZAEZF ZGBNHBMOEZHBPNBMAIJOBKL $. ${ $d A u x $. $d B u $. $d R u x $. $( Elementhood in a restricted domain quotient set. (Contributed by Peter Mazsa, 21-Aug-2020.) $) eldmqsres $p |- ( B e. V -> ( B e. ( dom ( R |` A ) /. ( R |` A ) ) <-> E. u e. A ( E. x x e. [ u ] R /\ B = [ u ] R ) ) ) $= ( wcel cres cdm cqs cv cec wceq wrex wex wa elqsg w3a 3bitr4i df-rex an12 wb cvv eldmres2 anbi1i ecres2 eqeq2d pm5.32i anbi2i 3ancoma df-3an 3anass elv 3bitr3i bitri exbii syl6bb ) DFGDECHZIZURJGDBKZURLZMZBUSNZAKUTELZGAOZ DVDMZPZBCNZBUSDURFQUTUSGZVBPZBOUTCGZVGPZBOVCVHVJVLBVJVKVEPZVBPZVLVIVMVBVI VMUBBACUTEUCUDUMUEVEVKVBPZPZVEVKVFPZPVNVLVOVQVEVKVBVFVKVAVDDCUTEUFUGUHUIV KVEVBRVEVKVBRVNVPVKVEVBUJVKVEVBUKVEVKVBULUNVKVEVFUASUOUPVBBUSTVGBCTSUQ $. $} ${ $d A u x $. $d B u x $. $d R u x $. $( Elementhood in a restricted domain quotient set. (Contributed by Peter Mazsa, 22-Aug-2020.) $) eldmqsres2 $p |- ( B e. V -> ( B e. ( dom ( R |` A ) /. ( R |` A ) ) <-> E. u e. A E. x e. [ u ] R B = [ u ] R ) ) $= ( wcel cres cdm cqs cv cec wex wceq wa wrex eldmqsres df-rex 19.41v bitri rexbii syl6bbr ) DFGDECHZIUCJGAKBKELZGZAMDUDNZOZBCPUFAUDPZBCPABCDEFQUHUGB CUHUEUFOAMUGUFAUDRUEUFASTUAUB $. $} ${ $d A x y $. $d B x y $. $d C x y $. $( Subclass theorem for quotient sets. (Contributed by Peter Mazsa, 12-Sep-2020.) $) qsss1 $p |- ( A C_ B -> ( A /. C ) C_ ( B /. C ) ) $= ( vy vx wss cv cec wceq wrex cab cqs ssrexv ss2abdv df-qs 3sstr4g ) ABFZD GEGCHIZEAJZDKREBJZDKACLBCLQSTDREABMNEDACOEDBCOP $. $} ${ qseq1i.1 $e |- A = B $. $( Equality theorem for quotient set, inference form. (Contributed by Peter Mazsa, 3-Jun-2021.) $) qseq1i $p |- ( A /. C ) = ( B /. C ) $= ( wceq cqs qseq1 ax-mp ) ABEACFBCFEDABCGH $. $} ${ qseq1d.1 $e |- ( ph -> A = B ) $. $( Equality theorem for quotient set, deduction form. (Contributed by Peter Mazsa, 27-May-2021.) $) qseq1d $p |- ( ph -> ( A /. C ) = ( B /. C ) ) $= ( wceq cqs qseq1 syl ) ABCFBDGCDGFEBCDHI $. $} ${ qseq2i.1 $e |- A = B $. $( Equality theorem for quotient set, inference form. (Contributed by Peter Mazsa, 3-Jun-2021.) $) qseq2i $p |- ( C /. A ) = ( C /. B ) $= ( wceq cqs qseq2 ax-mp ) ABECAFCBFEDABCGH $. $} ${ qseq2d.1 $e |- ( ph -> A = B ) $. $( Equality theorem for quotient set, deduction form. (Contributed by Peter Mazsa, 27-May-2021.) $) qseq2d $p |- ( ph -> ( C /. A ) = ( C /. B ) ) $= ( wceq cqs qseq2 syl ) ABCFDBGDCGFEBCDHI $. $} $( Equality theorem for quotient set. (Contributed by Peter Mazsa, 17-Apr-2019.) $) qseq12 $p |- ( ( A = B /\ C = D ) -> ( A /. C ) = ( B /. D ) ) $= ( wceq cqs qseq1 qseq2 sylan9eq ) ABECDEACFBCFBDFABCGCDBHI $. $( Binary relation on a restriction. (Contributed by Peter Mazsa, 2-Jan-2019.) $) brinxprnres $p |- ( C e. V -> ( B ( R i^i ( A X. ran ( R |` A ) ) ) C <-> ( B e. A /\ B R C ) ) ) $= ( cres crn cxp cin wbr wcel wa brres2 brresALTV syl5bbr ) BCDADAFZGHIJBCPJC EKBAKBCDJLABCDMABCDENO $. ${ $d A w x y z $. $d R w x y z $. $( Restriction of a class as a class of ordered pairs. (Contributed by Peter Mazsa, 2-Jan-2019.) $) inxprnres $p |- ( R i^i ( A X. ran ( R |` A ) ) ) = { <. x , y >. | ( x e. A /\ x R y ) } $= ( vz vw cres crn cxp cin cv wcel wbr wa copab wrel relxp wb cvv wceq el2v relin2 ax-mp relopab cop df-br brinxprnres elv bitr3i eleq1 breq1 anbi12d breq2 anbi2d opelopabg bitr4i eqrelriiv ) EFDCDCGHZIZJZAKZCLZVABKZDMZNZAB OZUSPUTPCURQDUSUBUCVEABUDEKZFKZUEZUTLZVGCLZVGVHDMZNZVIVFLZVJVGVHUTMZVMVGV HUTUFVOVMRFCVGVHDSUGUHUIVNVMREFVEVKVGVCDMZNVMABVGVHSSVAVGTVBVKVDVPVAVGCUJ VAVGVCDUKULVCVHTVPVLVKVCVHVGDUMUNUOUAUPUQ $. $} ${ $d A x y $. $d R x y $. $( Alternate definition of the restriction of a class. (Contributed by Peter Mazsa, 2-Jan-2019.) $) dfres4 $p |- ( R |` A ) = ( R i^i ( A X. ran ( R |` A ) ) ) $= ( vx vy cres cv wcel wbr wa copab crn cxp cin dfres2 inxprnres eqtr4i ) B AEZCFZAGRDFBHICDJBAQKLMCDABNCDABOP $. $} ${ $d A u $. $d B u $. $d V u $. $d W u $. $( Equivalent expressions with existential quantification. (Contributed by Peter Mazsa, 10-Sep-2021.) $) exan3 $p |- ( ( A e. V /\ B e. W ) -> ( E. u ( A e. [ u ] R /\ B e. [ u ] R ) <-> E. u ( u R A /\ u R B ) ) ) $= ( wcel wa cv cec wbr wb cvv elecALTV el2v1 bi2anan9 exbidv ) BEGZCFGZHBAI ZDJZGZCUAGZHTBDKZTCDKZHARUBUDSUCUERUBUDLATBDMENOSUCUELATCDMFNOPQ $. $} ${ $d B u $. $d C u $. $d V u $. $d W u $. $( Equivalent expressions with existential quantification. (Contributed by Peter Mazsa, 2-May-2021.) $) exanres $p |- ( ( B e. V /\ C e. W ) -> ( E. u ( u ( R |` A ) B /\ u ( S |` A ) C ) <-> E. u e. A ( u R B /\ u S C ) ) ) $= ( wcel wa cv cres wbr wex wrex brresALTV bi2anan9 anandi syl6bbr exbidv df-rex ) CGIZDHIZJZAKZCEBLMZUEDFBLMZJZANUEBIZUECEMZUEDFMZJZJZANULABOUDUHU MAUDUHUIUJJZUIUKJZJUMUBUFUNUCUGUOBUECEGPBUEDFHPQUIUJUKRSTULABUAS $. $} ${ $d B u $. $d C u $. $d V u $. $d W u $. $( Equivalent expressions with restricted existential quantification. (Contributed by Peter Mazsa, 10-Sep-2021.) $) exanres3 $p |- ( ( B e. V /\ C e. W ) -> ( E. u e. A ( B e. [ u ] R /\ C e. [ u ] S ) <-> E. u e. A ( u R B /\ u S C ) ) ) $= ( wcel wa cv cec wbr wb cvv elecALTV el2v1 bi2anan9 rexbidv ) CGIZDHIZJCA KZELIZDUBFLIZJUBCEMZUBDFMZJABTUCUEUAUDUFTUCUENAUBCEOGPQUAUDUFNAUBDFOHPQRS $. $} ${ $d B u $. $d C u $. $d V u $. $d W u $. $( Equivalent expressions with existential quantification. (Contributed by Peter Mazsa, 10-Sep-2021.) $) exanres2 $p |- ( ( B e. V /\ C e. W ) -> ( E. u ( u ( R |` A ) B /\ u ( S |` A ) C ) <-> E. u e. A ( B e. [ u ] R /\ C e. [ u ] S ) ) ) $= ( wcel wa cv cres wbr wex wrex cec exanres exanres3 bitr4d ) CGIDHIJAKZCE BLMTDFBLMJANTCEMTDFMJABOCTEPIDTFPIJABOABCDEFGHQABCDEFGHRS $. $} ${ $d A x y $. $( Restricted converse epsilon relation as a class of ordered pairs. (Contributed by Peter Mazsa, 10-Feb-2018.) $) cnvepres $p |- ( `' _E |` A ) = { <. x , y >. | ( x e. A /\ y e. x ) } $= ( cep ccnv cres cv wcel wbr wa copab dfres2 wb cvv brcnvep anbi2i opabbii elv eqtri ) DEZCFAGZCHZUABGZTIZJZABKUBUCUAHZJZABKABCTLUEUGABUDUFUBUDUFMAU AUCNORPQS $. $} $( Uniqueness condition for binary relationship over the ` 1st ` relationship (predates ~ br1steqg and has less conditions). (Contributed by Peter Mazsa, 30-Oct-2018.) $) br1steqORIG $p |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. 1st C <-> C = A ) ) $= ( wcel wa cop c1st cfv wceq wbr op1stg eqeq1d cvv wfn wb wfo fo1st fofn ax-mp opex fnbrfvb mp2an eqcom 3bitr3g ) ADFBEFGZABHZIJZCKZACKUHCILZCAKUGUI ACABDEMNIOPZUHOFUJUKQOOIRULSOOITUAABUBOUHCIUCUDACUEUF $. $( Uniqueness condition for binary relationship over the ` 2nd ` relationship (predates ~ br2ndeqg and has less conditions). (Contributed by Peter Mazsa, 30-Oct-2018.) $) br2ndeqORIG $p |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. 2nd C <-> C = B ) ) $= ( wcel wa cop c2nd cfv wceq wbr op2ndg eqeq1d cvv wfn wb wfo fo2nd fofn ax-mp opex fnbrfvb mp2an eqcom 3bitr3g ) ADFBEFGZABHZIJZCKZBCKUHCILZCBKUGUI BCABDEMNIOPZUHOFUJUKQOOIRULSOOITUAABUBOUHCIUCUDBCUEUF $. ${ $d A x y $. $d B x y $. $( Subclass relation in another form when the subclass is a relation. (Contributed by Peter Mazsa, 16-Feb-2019.) $) ssrel3 $p |- ( Rel A -> ( A C_ B <-> A. x A. y ( x A y -> x B y ) ) ) $= ( wrel wss cv cop wcel wi wal wbr ssrel df-br imbi12i 2albii syl6bbr ) CE CDFAGZBGZHZCIZTDIZJZBKAKRSCLZRSDLZJZBKAKABCDMUFUCABUDUAUEUBRSCNRSDNOPQ $. $} ${ $d A x y $. $d B x y $. $( Equality of relations. (Contributed by Peter Mazsa, 8-Mar-2019.) $) eqrel2 $p |- ( ( Rel A /\ Rel B ) -> ( A = B <-> A. x A. y ( x A y <-> x B y ) ) ) $= ( wrel wa wss cv wbr wi wal wceq wb ssrel3 bi2anan9 eqss 2albiim 3bitr4g ) CEZDEZFCDGZDCGZFAHZBHZCIZUCUDDIZJBKAKZUFUEJBKAKZFCDLUEUFMBKAKSUAUGTUBUH ABCDNABDCNOCDPUEUFABQR $. $} $( Intersection with a Cartesian product is a relation. (Contributed by Peter Mazsa, 4-Mar-2019.) $) relinxp $p |- Rel ( R i^i ( A X. B ) ) $= ( cxp wrel cin relxp relin2 ax-mp ) ABDZECJFEABGCJHI $. $( Range of converse is the domain. (Contributed by Peter Mazsa, 12-Feb-2018.) $) rncnv $p |- ran `' A = dom A $= ( cdm ccnv crn dfdm4 eqcomi ) ABACDAEF $. ${ $d R x $. $( Alternate definition of domain. (Contributed by Peter Mazsa, 2-Mar-2018.) $) dfdm6 $p |- dom R = { x | [ x ] R =/= (/) } $= ( cv cec c0 wne cdm ecdmn0 abbi2i ) ACZBDEFABGJBHI $. $} ${ $d R x $. $( Alternate definition of range. (Contributed by Peter Mazsa, 1-Aug-2018.) $) dfrn6 $p |- ran R = { x | [ x ] `' R =/= (/) } $= ( crn ccnv cdm cv cec c0 wne cab df-rn dfdm6 eqtri ) BCBDZEAFNGHIAJBKANLM $. $} ${ $d A x y $. $( The range of the restricted converse epsilon is the union of the restriction. (Contributed by Peter Mazsa, 11-Feb-2018.) (Revised by Peter Mazsa, 26-Sep-2021.) $) rncnvepres $p |- ran ( `' _E |` A ) = U. A $= ( vx vy cv wcel wa copab crn wex cab ccnv cres cuni rnopab cnvepres rneqi cep wrex dfuni2 df-rex abbii eqtri 3eqtr4i ) BDZAECDUDEZFZBCGZHUFBIZCJZQK ALZHAMZUFBCNUJUGBCAOPUKUEBARZCJUICBASULUHCUEBATUAUBUC $. $} ${ dmecd.1 $e |- ( ph -> dom R = A ) $. dmecd.2 $e |- ( ph -> [ B ] R = [ C ] R ) $. $( Equality of the coset of ` B ` and the coset of ` C ` implies equivalence of domain elementhood (equivalence is not necessary as opposed to ~ ereldm ). (Contributed by Peter Mazsa, 9-Oct-2018.) $) dmecd $p |- ( ph -> ( B e. A <-> C e. A ) ) $= ( cdm wcel cec c0 wne neeq1d ecdmn0 3bitr4g eleq2d 3bitr3d ) ACEHZIZDRIZC BIDBIACEJZKLDEJZKLSTAUAUBKGMCENDENOARBCFPARBDFPQ $. $} ${ dmec2d.1 $e |- ( ph -> [ B ] R = [ C ] R ) $. $( Equality of the coset of ` B ` and the coset of ` C ` implies equivalence of domain elementhood (equivalence is not necessary as opposed to ~ ereldm ). (Contributed by Peter Mazsa, 12-Oct-2018.) $) dmec2d $p |- ( ph -> ( B e. dom R <-> C e. dom R ) ) $= ( cdm eqidd dmecd ) ADFZBCDAIGEH $. $} $( Intersection with a Cartesian product is a subclass of restriction. (Contributed by Peter Mazsa, 19-Jul-2019.) $) inxpssres $p |- ( R i^i ( A X. B ) ) C_ ( R |` A ) $= ( cxp cin cvv cres wss ssid ssv xpss12 mp2an sslin ax-mp df-res sseqtr4i ) CABDZEZCAFDZEZCAGQSHZRTHAAHBFHUAAIBJAABFKLQSCMNCAOP $. $( Property of the identity binary relation. (Contributed by Peter Mazsa, 18-Dec-2021.) $) brid $p |- ( A _I B <-> B _I A ) $= ( cid wbr ccnv cnvi breqi wrel wb reli relbrcnvg ax-mp bitr3i ) ABCDABCEZDZ BACDZABNCFGCHOPIJABCKLM $. $( For sets, the identity binary relation is the same as equality. (Contributed by Peter Mazsa, 24-Jun-2020.) (Revised by Peter Mazsa, 18-Dec-2021.) $) ideq2 $p |- ( A e. V -> ( A _I B <-> A = B ) ) $= ( cid wbr wcel wceq brid ideqg eqcom syl6bb syl5bb ) ABDEBADEZACFZABGZABHNM BAGOBACIBAJKL $. $( Condition for the identity restriction to be a subclass of identity intersection with a Cartesian product. (Contributed by Peter Mazsa, 19-Jul-2018.) $) idresssidinxp $p |- ( A C_ B -> ( _I |` A ) C_ ( _I i^i ( A X. B ) ) ) $= ( wss cid cres cxp resss a1i idssxp xpss2 syl5ss ssind ) ABCZDAEZDABFZNDCMD AGHMNAAFOAIABAJKL $. $( Condition for the identity restriction to be equal to the identity intersection with a Cartesian product. (Contributed by Peter Mazsa, 19-Jul-2018.) $) idreseqidinxp $p |- ( A C_ B -> ( _I i^i ( A X. B ) ) = ( _I |` A ) ) $= ( wss cid cxp cin cres inxpssres a1i idresssidinxp eqssd ) ABCZDABEFZDAGZMN CLABDHIABJK $. $( Property of identity relation, cf. ~ extep , ~ extssr and the comment of ~ df-ssr . (Contributed by Peter Mazsa, 5-Jul-2019.) $) extid $p |- ( A e. V -> ( [ A ] `' _I = [ B ] `' _I <-> A = B ) ) $= ( cid ccnv cec wceq csn wcel cnvi eceq2i ecidsn eqtri eqeq12i sneqbg syl5bb ) ADEZFZBQFZGAHZBHZGACIABGRTSUARADFTQDAJKALMSBDFUAQDBJKBLMNABCOP $. ${ $d A x y $. $d B x y $. $d R x y $. $d S x y $. $( Two ways to say that an intersection with a Cartesian product is a subclass. (Contributed by Peter Mazsa, 16-Jul-2019.) $) inxpss $p |- ( ( R i^i ( A X. B ) ) C_ S <-> A. x e. A A. y e. B ( x R y -> x S y ) ) $= ( cv cxp cin wbr wi wal wcel wa wss wral brinxp2ALTV imbi1i impexp bitri 2albii wrel wb relinxp ssrel3 ax-mp r2al 3bitr4i ) AGZBGZECDHIZJZUIUJFJZK ZBLALZUICMUJDMNZUIUJEJZUMKZKZBLALUKFOZURBDPACPUNUSABUNUPUQNZUMKUSULVAUMCD UIUJEQRUPUQUMSTUAUKUBUTUOUCCDEUDABUKFUEUFURABCDUGUH $. $} ${ $d A x y $. $d B x y $. $d R x y $. $( Two ways to say that an intersection of the identity relation with a Cartesian product is a subclass. (Contributed by Peter Mazsa, 16-Jul-2019.) $) idinxpss $p |- ( ( _I i^i ( A X. B ) ) C_ R <-> A. x e. A A. y e. B ( x = y -> x R y ) ) $= ( cid cxp cin wss cv wbr wi wral wceq inxpss wb cvv ideqg elv imbi1i 2ralbii bitri ) FCDGHEIAJZBJZFKZUCUDEKZLZBDMACMUCUDNZUFLZBDMACMABCDFEOUGU IABCDUEUHUFUEUHPBUCUDQRSTUAUB $. $} ${ $d x y $. $d A y $. $( Two ways to say that an intersection with a Cartesian product is a subclass (cf. ~ inxpss ). (Contributed by Peter Mazsa, 8-Mar-2019.) $) inxpss3 $p |- ( A. x A. y ( x ( R i^i ( A X. B ) ) y -> x ( S i^i ( A X. B ) ) y ) <-> A. x e. A A. y e. B ( x R y -> x S y ) ) $= ( cv cxp cin wbr wal wcel wral brinxp2ALTV imbi12i imdistan bitr4i 2albii wi wa r2al ) AGZBGZECDHZIJZUBUCFUDIJZSZBKAKUBCLUCDLTZUBUCEJZUBUCFJZSZSZBK AKUKBDMACMUGULABUGUHUITZUHUJTZSULUEUMUFUNCDUBUCENCDUBUCFNOUHUIUJPQRUKABCD UAQ $. $} ${ $d A x y $. $d B x y $. $d R x y $. $d S x y $. $( Two ways to say that intersections with Cartesian products are in a subclass relation. (Contributed by Peter Mazsa, 8-Mar-2019.) $) inxpss2 $p |- ( ( R i^i ( A X. B ) ) C_ ( S i^i ( A X. B ) ) <-> A. x e. A A. y e. B ( x R y -> x S y ) ) $= ( cxp cin wss cv wbr wi wal wral wrel wb relinxp ssrel3 ax-mp inxpss3 bitri ) ECDGZHZFUBHZIZAJZBJZUCKUFUGUDKLBMAMZUFUGEKUFUGFKLBDNACNUCOUEUHPCD EQABUCUDRSABCDEFTUA $. $} ${ $d A x y $. $d B x y $. $d R x y $. $( Two ways to say that intersections with Cartesian products are in a subclass relation, special case of ~ inxpss2 . (Contributed by Peter Mazsa, 4-Jul-2019.) $) inxpssidinxp $p |- ( ( R i^i ( A X. B ) ) C_ ( _I i^i ( A X. B ) ) <-> A. x e. A A. y e. B ( x R y -> x = y ) ) $= ( cxp cin cid wss cv wbr wi wral wceq inxpss2 wb cvv ideqg elv imbi2i 2ralbii bitri ) ECDFZGHUCGIAJZBJZEKZUDUEHKZLZBDMACMUFUDUENZLZBDMACMABCDEH OUHUJABCDUGUIUFUGUIPBUDUEQRSTUAUB $. $} ${ $d A x y $. $d B x y $. $d R x y $. $( Two ways to say that intersections with Cartesian products are in a subclass relation, special case of ~ inxpss2 . (Contributed by Peter Mazsa, 6-Mar-2019.) $) idinxpssinxp $p |- ( ( _I i^i ( A X. B ) ) C_ ( R i^i ( A X. B ) ) <-> A. x e. A A. y e. B ( x = y -> x R y ) ) $= ( cid cxp cin wss cv wbr wi wral wceq inxpss2 wb cvv ideqg elv imbi1i 2ralbii bitri ) FCDGZHEUCHIAJZBJZFKZUDUEEKZLZBDMACMUDUENZUGLZBDMACMABCDFE OUHUJABCDUFUIUGUFUIPBUDUEQRSTUAUB $. $} $( The intersection of the identity function with a square cross product. (Contributed by FL, 2-Aug-2009.) $) residcp $p |- ( _I i^i ( A X. A ) ) = ( _I |` A ) $= ( cid cxp cin cres cvv wss ssid ssv xpss12 mp2an sslin ax-mp sseqtr4i resss df-res idssxp ssini eqssi ) BAACZDZBAEZUABAFCZDZUBTUCGZUAUDGAAGAFGUEAHAIAAA FJKTUCBLMBAPNUBBTBAOAQRS $. ${ $d A x $. $d R x $. $( Identity intersection with a square Cartesian product in subclass relation with an intersection with the same Cartesian product. (Contributed by Peter Mazsa, 4-Mar-2019.) $) idinxpssinxp2 $p |- ( ( _I i^i ( A X. A ) ) C_ ( R i^i ( A X. A ) ) <-> A. x e. A x R x ) $= ( cid cxp cin wss cv wcel wbr wral cres residcp sseq1i issref brinxp2ALTV wa pm4.24 anbi1i bitr4i ralbii 3bitri ralanid bitri ) DBBEZFZCUEFZGZAHZBI ZUIUICJZQZABKZUKABKUHDBLZUGGUIUIUGJZABKUMUFUNUGBMNABUGOUOULABUOUJUJQZUKQU LBBUIUICPUJUPUKUJRSTUAUBUKABUCUD $. $} ${ $d A x $. $d R x $. $( Identity intersection with a square Cartesian product in subclass relation with an intersection with the same Cartesian product. (Contributed by Peter Mazsa, 16-Mar-2019.) $) idinxpssinxp3 $p |- ( ( _I i^i ( A X. A ) ) C_ ( R i^i ( A X. A ) ) <-> ( _I |` A ) C_ R ) $= ( vx cid cxp cin wss cv wbr wral cres idinxpssinxp2 issref bitr4i ) DAAEZ FBOFGCHZPBICAJDAKBGCABLCABMN $. $} ${ $d A x y $. $d R x y $. $( Identity intersection with a square Cartesian product in subclass relation with an intersection with the same Cartesian product (cf. ~ idinxpssinxp2 ). (Contributed by Peter Mazsa, 8-Mar-2019.) $) idinxpssinxp4 $p |- ( A. x e. A A. y e. A ( x = y -> x R y ) <-> A. x e. A x R x ) $= ( cv wceq wbr wi wral cid cxp cin wss idinxpssinxp idinxpssinxp2 bitr3i ) AEZBEZFQRDGHBCIACIJCCKZLDSLMQQDGACIABCCDNACDOP $. $} ${ $d R x y $. $( Two ways of saying a relation is symmetric. (Contributed by FL, 31-Aug-2009.) $) twsymr $p |- ( Rel R -> ( R = `' R <-> A. x A. y ( x R y -> y R x ) ) ) $= ( ccnv wceq wss wa wrel cv wbr wi wal eqss cnvsym biimpi a1d adantl com12 dfrel2 cnvss sseq1 syl5ibcom sylbir sylbi imp biimpri jca impbid syl5bb ex ) CCDZECUKFZUKCFZGZCHZAIZBIZCJUQUPCJKBLALZCUKMUOUNURUNUOURUMUOURKULUMU RUOUMURABCNZOPQRUOURUNUOURGULUMUOURULUOUKDZCEZURULKCSURVAULURUMVAULKUSUMU TUKFVAULUKCTUTCUKUAUBUCRUDUEURUMUOUMURUSUFQUGUJUHUI $. $} ${ $d R x y $. $( Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 23-Aug-2018.) $) relcnveq $p |- ( Rel R -> ( `' R C_ R <-> `' R = R ) ) $= ( vx vy ccnv wceq wrel wss eqcom cv wbr wi twsymr cnvsym syl6bbr syl5rbbr wal ) ADZAEAQEZAFZQAGZAQHSRBIZCIZAJUBUAAJKCPBPTBCALBCAMNO $. $} ${ $d R x y $. $( Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 28-Apr-2019.) $) relcnveq2 $p |- ( Rel R -> ( `' R = R <-> A. x A. y ( x R y <-> y R x ) ) ) $= ( wrel ccnv wss wa cv wbr wi wceq wb cnvsym dfrel2 biimpi sseq1d syl5rbbr wal a1i relbrcnvg imbi12d 2albidv bitrd anbi12d eqss 2albiim 3bitr4g ) CD ZCEZCFZCUIFZGAHZBHZCIZUMULCIZJBRARZUOUNJZBRARZGUICKUNUOLBRARUHUJUPUKURUJU PLUHABCMSUHUKULUMUIIZUMULUIIZJZBRARZURVBUIEZUIFUHUKABUIMUHVCCUIUHVCCKCNOP QUHVAUQABUHUSUOUTUNULUMCTUMULCTUAUBUCUDUICUEUNUOABUFUG $. $} ${ $d R x y $. $( Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 28-Apr-2019.) $) relcnveq3 $p |- ( Rel R -> ( `' R C_ R <-> A. x A. y ( x R y <-> y R x ) ) ) $= ( wrel ccnv wss wceq cv wbr wb wal relcnveq relcnveq2 bitrd ) CDCEZCFOCGA HZBHZCIQPCIJBKAKCLABCMN $. $} ${ $d A u v $. $d R u v $. $( Simplification of a special quotient set. (Contributed by Peter Mazsa, 2-Sep-2020.) $) qsresid $p |- ( A /. ( R |` A ) ) = ( A /. R ) $= ( vu vv cv cres cec wceq wrex cab cqs wcel wa ecres2 eqeq2d pm5.32i exbii wex df-rex df-qs 3bitr4i abbii 3eqtr4i ) CEZDEZBAFZGZHZDAIZCJUDUEBGZHZDAI ZCJAUFKABKUIULCUEALZUHMZDRUMUKMZDRUIULUNUODUMUHUKUMUGUJUDAUEBNOPQUHDASUKD ASUAUBDCAUFTDCABTUC $. $} $( The empty set has no elements. (Contributed by Peter Mazsa, 4-Jan-2018.) $) nel02 $p |- ( A = (/) -> -. B e. A ) $= ( c0 wceq wcel wn noel eleq2 notbid mpbiri ) ACDZBAEZFBCEZFBGKLMACBHIJ $. ${ $d A x $. $d R x $. $( Two ways of expressing that the empty set is not an element of a quotient set. (Contributed by Peter Mazsa, 5-Dec-2019.) $) n0elqs $p |- ( -. (/) e. ( A /. R ) <-> A C_ dom R ) $= ( vx cv cdm wcel wral cec c0 wne wss cqs wn ecdmn0 wrex rexbii notbii cvv wceq 3bitr4ri ralbii dfss3 nne dfral2 wb 0ex elqsg ax-mp eqcom bitri ) CD ZBEZFZCAGUKBHZIJZCAGZAULKIABLFZMZUMUOCAUKBNUACAULUBUOMZCAOZMUNISZCAOZMUPU RUTVBUSVACAUNIUCPQUOCAUDUQVBUQIUNSZCAOZVBIRFUQVDUEUFCAIBRUGUHVCVACAIUNUIP UJQTT $. $} $( Two ways of expressing that the empty set is not an element of a quotient set. (Contributed by Peter Mazsa, 25-Jul-2021.) $) n0elqs2 $p |- ( -. (/) e. ( A /. R ) <-> dom ( R |` A ) = A ) $= ( c0 cqs wcel wn cdm wss cres wceq n0elqs ssdmres bitri ) CABDEFABGHBAIGAJA BKABLM $. $( Condition for a coset to be a set. (Contributed by Peter Mazsa, 4-May-2019.) $) ecex2 $p |- ( ( R |` A ) e. V -> ( B e. A -> [ B ] R e. _V ) ) $= ( wcel cres cec cvv ecexg ecres2 eleq1d syl5ib com12 ) BAEZCAFZDEZBCGZHEZPB OGZHENRBDOINSQHABCJKLM $. ${ $d A x y $. $d R x y $. $d V x $. $( The union of a quotient set: a weaker version of ~ uniqs . (Contributed by Peter Mazsa, 20-Jun-2019.) $) uniqsALTV $p |- ( ( R |` A ) e. V -> U. ( A /. R ) = ( R " A ) ) $= ( vy vx cres wcel cec wceq wrex cab cuni ciun cqs cima cvv ecex2 ralrimiv cv wral dfiun2g syl eqcomd df-qs unieqi csn df-ec iuneq2i imaiun 3eqtr2ri a1i iunid imaeq2i 3eqtr4g ) BAFCGZDSESZBHZIEAJDKZLZEAUQMZABNZLBAOZUOUTUSU OUQPGZEATUTUSIUOVCEAAUPBCQREDAUQPUAUBUCVAUREDABUDUEUTEABUPUFZOZMBEAVDMZOV BEAUQVEUQVEIUPAGUPBUGUKUHEBAVDUIVFABEAULUMUJUN $. $} $( The range of a restriction is equal to the union of the quotient set. (Contributed by Peter Mazsa, 19-May-2018.) $) rnresequniqs $p |- ( ( R |` A ) e. V -> ran ( R |` A ) = U. ( A /. R ) ) $= ( cres wcel cqs cuni cima crn uniqsALTV df-ima syl6req ) BADZCEABFGBAHMIABC JBAKL $. ${ $d A x y $. $( Two ways of expressing that the empty set is not an element of a class. (Contributed by Peter Mazsa, 31-Jan-2018.) $) n0el2 $p |- ( -. (/) e. A <-> dom ( `' _E |` A ) = A ) $= ( vx vy c0 wcel wn cv wa copab cdm wceq cep ccnv cres wex wral n0el bitri dmopab3 cnvepres dmeqi eqeq1i bitr4i ) DAEFZBGZAECGUEEZHBCIZJZAKZLMANZJZA KUDUFCOBAPUIBCAQUFBCASRUKUHAUJUGBCATUAUBUC $. $} ${ $d A x y $. $d V x $. $( Sethood condition for the restricted converse epsilon relation. (Contributed by Peter Mazsa, 24-Sep-2018.) $) cnvepresex $p |- ( A e. V -> ( `' _E |` A ) e. _V ) $= ( vx vy wcel cv wa copab cvv cep ccnv cres cab vex abid2 eleq1i mpbir a1i elex adantl opabex3d cnvepres sylibr ) ABEZCFZAEZDFUEEZGCDHZIEJKALZIEUDUG CDAABSUFUGDMZIEZUDUKUFUKUEIECNUJUEIDUEOPQRTUAUIUHICDAUBPUC $. $} $( Sethood condition for the intersection relation, cf. ~ inex1g . (Contributed by Peter Mazsa, 19-Dec-2018.) $) inex2ALTV $p |- ( A e. V -> ( B i^i A ) e. _V ) $= ( wcel cin cvv incom inex1g syl5eqel ) ACDBAEABEFBAGABCHI $. $( More general sethood condition for the intersection relation. (Contributed by Peter Mazsa, 24-Nov-2019.) $) inex3 $p |- ( ( A e. V \/ B e. W ) -> ( A i^i B ) e. _V ) $= ( wcel cin cvv inex1g inex2ALTV jaoi ) ACEABFGEBDEABCHBADIJ $. $( Sethood condition for the intersection with a Cartesian product. (Contributed by Peter Mazsa, 10-May-2019.) $) inxpex $p |- ( ( R e. W \/ ( A e. U /\ B e. V ) ) -> ( R i^i ( A X. B ) ) e. _V ) $= ( wcel cxp cin cvv wa inex1g xpexg inex2ALTV syl jaoi ) CFGCABHZIJGZADGBEGK ZCQFLSQJGRABDEMQCJNOP $. ${ eqres.1 $e |- R = ( S |` C ) $. $( Converting a class constant definition by restriction (like ~ df-ers or ~ df-parts ) into a binary relation. (Contributed by Peter Mazsa, 1-Oct-2018.) $) eqres $p |- ( B e. V -> ( A R B <-> ( A e. C /\ A S B ) ) ) $= ( wbr cres wcel wa breqi brresALTV syl5bb ) ABDHABECIZHBFJACJABEHKABDOGLC ABEFMN $. $} $( The ordered pair ` <. A , A >. ` in Kuratowski's representation (predates ~ opidg ). (Contributed by Peter Mazsa, 22-Jul-2019.) $) opidORIG $p |- ( A e. V -> <. A , A >. = { { A } } ) $= ( wcel cop csn cpr wceq dfopg anidms dfsn2 eqcomi preq2i eqtr4i syl6eq ) AB CZAADZAEZAAFZFZQEZOPSGAABBHISQQFTRQQQRAJKLQJMN $. $( Equality conditions for ordered pairs ` <. A , A >. ` and ` <. B , B >. ` . (Contributed by Peter Mazsa, 22-Jul-2019.) $) opideq $p |- ( ( A e. V /\ B e. W ) -> ( <. A , A >. = <. B , B >. <-> A = B ) ) $= ( wcel wa cop wceq csn opidORIG eqeqan12d wb cvv sneqbg ax-mp syl5bb adantr snexALT bitrd ) ACEZBDEZFAAGZBBGZHAIZIZBIZIZHZABHZTUAUBUEUCUGACJBDJKTUHUILU AUHUDUFHZTUIUDMEUHUJLARUDUFMNOABCNPQS $. $( Ordered pair element in an intersection with Cartesian product. (Contributed by Peter Mazsa, 21-Jul-2019.) $) opelinxp $p |- ( <. C , D >. e. ( R i^i ( A X. B ) ) <-> ( ( C e. A /\ D e. B ) /\ <. C , D >. e. R ) ) $= ( cxp cin wbr wcel wa cop brinxp2ALTV df-br anbi2i 3bitr3i ) CDEABFGZHCAIDB IJZCDEHZJCDKZPIQSEIZJABCDELCDPMRTQCDEMNO $. ${ $d A x y $. $( A subclass of the identity relation is the intersection of identity relation with Cartesian product of the domain and range of the class. (Contributed by Peter Mazsa, 22-Jul-2019.) $) iss2 $p |- ( A C_ _I <-> A = ( _I i^i ( dom A X. ran A ) ) ) $= ( vx vy cid wss wceq cv cop wcel wb wal wa wi vex a1i jcad syl6ibr syl5bi wex wrel cdm crn cxp cin ssel opeldm opelrn anandi wbr df-br bitr3i eldm2 ideq opeq2 eleq1d biimprcd sylcom exlimdv imbi2d syl5ibcom imp cvv opideq elrn2 el2v biimpri eqtr3d anim12d pm4.24 ex impd impbid opelinxp alrimivv bian syl6bbr relss relinxp eqrel sylancl mpbird inss1 sseq1 mpbiri impbii reli mpi ) ADEZADAUAZAUBZUCZUDZFZWHWMBGZCGZHZAIZWPWLIZJZCKBKZWHWSBCWHWQWP DIZWNWIIZWOWJIZLZLZWRWHWQXEWHWQXAXBLZXAXCLZLXEWHWQXFXGWHWQXAXBADWPUEZWQXB MWHWNWOABNZCNZUFOPWHWQXAXCXHWQXCMWHWNWOAXIXJUGOPPXAXBXCUHQWHXAXDWQXAWNWOF ZWHXDWQMZXAWNWODUIXKWNWODUJWNWOXJUMUKZWHXKXLWHXKLZXDWQWQLWQXNXBWQXCWQWHXK XBWQMZWHXBWNWNHZAIZMXKXOXBWQCSWHXQCWNAXIULWHWQXQCWHWQXAXQXHXAXKWQXQXMXKXQ WQXKXPWPAWNWOWNUNZUOZUPRUQURRXKXQWQXBXSUSUTVAWHXKXCWQMZWHXCWOWOHZAIZMXKXT XCWQBSWHYBBWOAXJVDWHWQYBBWHWQXAYBXHXAXKWQYBXMXKYBWQXKYAWPAXKXPYAWPXPYAFZX KYCXKJBCWNWOVBVBVCVEVFXRVGUOZUPRUQURRXKYBWQXCYDUSUTVAVHWQVIQVJRVKVLWRXAXD WIWJWNWODVMVOVPVNWHATZWLTWMWTJWHDTYEWFADVQWGWIWJDVRBCAWLVSVTWAWMWHWLDEDWK WBAWLDWCWDWE $. $} ${ $d A u $. $d R u $. $d V u $. $( Elementhood in a domain of a converse. (Contributed by Peter Mazsa, 25-May-2018.) $) eldmcnv $p |- ( A e. V -> ( A e. dom `' R <-> E. u u R A ) ) $= ( wcel ccnv cdm cv wbr wex eldmg cvv wb vex brcnvg mpan2 exbidv bitrd ) B DEZBCFZGEBAHZTIZAJUABCIZAJABTDKSUBUCASUALEUBUCMANBUADLCOPQR $. $} $( Alternate definition of the relation predicate. (Contributed by Peter Mazsa, 6-Nov-2018.) $) dfrel5 $p |- ( Rel R <-> ( R |` dom R ) = R ) $= ( wrel ccnv wceq cdm cres dfrel2 resdm2 eqeq1i bitr4i ) ABACCZADAAEFZADAGLK AAHIJ $. $( Alternate definition of the relation predicate. (Contributed by Peter Mazsa, 14-Mar-2019.) $) dfrel6 $p |- ( Rel R <-> ( R i^i ( dom R X. ran R ) ) = R ) $= ( wrel cdm cres wceq crn cxp cin dfrel5 dfres3 eqeq1i bitri ) ABAACZDZAEAMA FGHZAEAINOAAMJKL $. $( Converse restricted to range is converse. (Contributed by Peter Mazsa, 3-Sep-2021.) $) cnvresrn $p |- ( `' R |` ran R ) = `' R $= ( ccnv crn cres cdm df-rn reseq2i wrel wceq relcnv dfrel5 mpbi eqtri ) ABZA CZDNNEZDZNOPNAFGNHQNIAJNKLM $. ${ $d A x $. $d B x $. $d R x $. $d V x $. $d W x $. $( Two ways of saying that the coset of ` A ` and the coset of ` B ` have no elements in common. (Contributed by Peter Mazsa, 1-Dec-2018.) $) ecin0 $p |- ( ( A e. V /\ B e. W ) -> ( ( [ A ] R i^i [ B ] R ) = (/) <-> A. x ( A R x -> -. B R x ) ) ) $= ( cec cin c0 wceq cv wcel wn wi wal wa wbr disj1 wb cvv elecg el2v1 el2v2 adantr elecALTV adantl notbid imbi12d albidv syl5bb ) BDGZCDGZHIJAKZUKLZU MULLZMZNZAOBELZCFLZPZBUMDQZCUMDQZMZNZAOAUKULRUTUQVDAUTUNVAUPVCURUNVASZUSU RVEAUMBDTEUAUBUDUTUOVBUSUOVBSZURUSVFACUMDFTUEUCUFUGUHUIUJ $. $} ${ $d A x $. $d B x $. $d R x $. $d V x $. $d W x $. $( Two ways of saying that the coset of ` A ` and the coset of ` B ` have some elements in common. (Contributed by Peter Mazsa, 23-Jan-2019.) $) ecinn0 $p |- ( ( A e. V /\ B e. W ) -> ( ( [ A ] R i^i [ B ] R ) =/= (/) <-> E. x ( A R x /\ B R x ) ) ) $= ( wcel wa cec cin c0 wne cv wbr wn wi wal wex ecin0 necon3abid notnotb anbi2i exbii exanali bitri syl6bbr ) BEGCFGHZBDICDIJZKLBAMZDNZCUIDNZOZPAQ ZOZUJUKHZARZUGUMUHKABCDEFSTUPUJULOZHZARUNUOURAUKUQUJUKUAUBUCUJULAUDUEUF $. $} ${ $d B z $. $d C z $. $d D z $. $d x z $. $d y z $. $( Lemma for ~ inecmo . (Contributed by Peter Mazsa, 29-May-2018.) $) ineleq $p |- ( A. x e. A A. y e. B ( x = y \/ ( C i^i D ) = (/) ) <-> A. x e. A A. z A. y e. B ( ( z e. C /\ z e. D ) -> x = y ) ) $= ( cv wceq cin c0 wo wral wcel wa wi wal wex bitri ralbii orcom df-or neq0 wn elin exbii imbi1i 19.23v bitr4i 3bitri ralcom4 ) AHBHIZFGJZKIZLZBEMZCH ZFNUQGNOZULPZBEMCQZADUPUSCQZBEMUTUOVABEUOUNULLUNUDZULPZVAULUNUAUNULUBVCUR CRZULPVAVBVDULVBUQUMNZCRVDCUMUCVEURCUQFGUEUFSUGURULCUHUIUJTUSBCEUKST $. $} ${ $d A x y z $. $d B y z $. $d C x z $. $d R x y z $. inecmo.1 $e |- ( x = y -> B = C ) $. $( Lemma for dfeldisj5 (via ~ inecmo2 ), ~ dfdisjs5 , ~ dfdisjALTV5 , ~ eldisjs5 (via ~ inecmo3 , ~ cosscnvssid5 ), ~ dffunsALTV5 (via ~ ineccnvmo , ~ ineccnvmo2 ), and ~ dffunALTV5 (via ~ cossssid5 ). (Contributed by Peter Mazsa, 29-May-2018.) $) inecmo $p |- ( Rel R -> ( A. x e. A A. y e. A ( x = y \/ ( [ B ] R i^i [ C ] R ) = (/) ) <-> A. z E* x e. A B R z ) ) $= ( wrel cv wbr wal cec wcel wa wceq wi wral relelec syl6rbbr c0 wo anbi12d wrmo cin imbi1d 2ralbidv breq1d rmo4 albidv ineleq ralcom4 bitri ) GIZECJ ZGKZADUDZCLUOEGMZNZUOFGMZNZOZAJBJPZQZBDRZADRZCLZVCURUTUEUAPUBBDRADRZUNUQV FCUNVFUPFUOGKZOZVCQZBDRADRUQUNVDVKABDDUNVBVJVCUNUSUPVAVIUOEGSUOFGSUCUFUGU PVIABDVCEFUOGHUHUITUJVHVECLADRVGABCDDURUTUKVEACDULUMT $. $} ${ $d A u v x $. $d B u v x $. $d R u v x $. $( Lemma for dfeldisj5 , and for ~ dfdisjs5 , ~ dfdisjALTV5 , ~ eldisjs5 (via ~ inecmo3 , ~ cosscnvssid5 ). (Contributed by Peter Mazsa, 29-May-2018.) (Revised by Peter Mazsa, 2-Sep-2021.) $) inecmo2 $p |- ( ( A. u e. A A. v e. A ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) /\ Rel R ) <-> ( A. x E* u e. A u R x /\ Rel R ) ) $= ( wrel cv wceq cec cin c0 wo wral wbr wrmo wal id inecmo pm5.32ri ) EFCGZ BGZHZTEIUAEIJKHLBDMCDMTAGENCDOAPCBADTUAEUBQRS $. $} ${ $d B x y z $. $d F x y z $. $( Lemma for ~ ineccnvmo2 . (Contributed by Peter Mazsa, 2-Sep-2021.) $) ineccnvmo $p |- ( A. y e. B A. z e. B ( y = z \/ ( [ y ] `' F i^i [ z ] `' F ) = (/) ) <-> A. x E* y e. B x F y ) $= ( cv wceq ccnv cec cin c0 wo wral wbr wrmo wal wrel wb relcnv cvv id el2v inecmo ax-mp brcnvg rmobii albii bitri ) BFZCFZGZUIEHZIUJULIJKGLCDMBDMZUI AFZULNZBDOZAPZUNUIENZBDOZAPULQUMUQRESBCADUIUJULUKUAUCUDUPUSAUOURBDUOURRBA UIUNTTEUEUBUFUGUH $. $} $( Lemma for ~ ineccnvmo2 . (Contributed by Peter Mazsa, 3-Sep-2021.) $) alrmomo $p |- ( A. x E* y e. ran R x R y <-> A. x E* y x R y ) $= ( cv wbr crn wrmo wmo wcel wa df-rmo ccnv cres cnvresrn breqi cvv brresALTV wb elv bitri brcnvg el2v anbi2i 3bitr3i mobii albii ) ADZBDZCEZBCFZGZUIBHZA UKUHUJIZUIJZBHULUIBUJKUNUIBUHUGCLZUJMZEZUHUGUOEZUNUIUHUGUPUOCNOUQUMURJZUNUQ USRAUJUHUGUOPQSURUIUMURUIRBAUHUGPPCUAUBZUCTUTUDUETUF $. ${ $d R u $. $d R x $. $( Lemma for ~ inecmo3 . (Contributed by Peter Mazsa, 5-Sep-2021.) $) alrmomo2 $p |- ( Rel R -> ( A. x E* u e. dom R u R x <-> A. x E* u u R x ) ) $= ( wrel cv wbr cdm wrmo wmo wcel df-rmo cres cvv brresALTV elv wceq dfrel5 wa wb biimpi breqd syl5bbr mobidv syl5bb albidv ) CDZBEZAEZCFZBCGZHZUIBIZ AUKUGUJJUIRZBIUFULUIBUJKUFUMUIBUMUGUHCUJLZFZUFUIUOUMSAUJUGUHCMNOUFUNCUGUH UFUNCPCQTUAUBUCUDUE $. $} ${ $d F u x y $. $( Lemma for ~ dffunsALTV5 and ~ dffunALTV5 (via ~ cossssid5 ). (Contributed by Peter Mazsa, 4-Sep-2021.) $) ineccnvmo2 $p |- ( A. x e. ran F A. y e. ran F ( x = y \/ ( [ x ] `' F i^i [ y ] `' F ) = (/) ) <-> A. u E* x u F x ) $= ( cv wceq ccnv cec cin c0 wo crn wral wbr wal wmo ineccnvmo alrmomo bitri wrmo ) AEZBEZFUADGZHUBUCHIJFKBDLZMAUDMCEUADNZAUDTCOUEAPCOCABUDDQCADRS $. $} ${ $d R u v x $. $( Lemma for ~ dfdisjs5 , ~ dfdisjALTV5 , ~ eldisjs5 (via ~ cosscnvssid5 ). (Contributed by Peter Mazsa, 5-Sep-2021.) $) inecmo3 $p |- ( ( A. u e. dom R A. v e. dom R ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) /\ Rel R ) <-> ( A. x E* u u R x /\ Rel R ) ) $= ( cv wceq cec cin c0 wo cdm wral wrel wbr wrmo wal wmo inecmo2 alrmomo2 wa pm5.32ri bitri ) CEZBEZFUCDGUDDGHIFJBDKZLCUELDMZTUCAEDNZCUEOAPZUFTUGCQ APZUFTABCUEDRUFUHUIACDSUAUB $. $} ${ bropabid.1 $e |- R = { <. x , y >. | ph } $. $( Lemma for ~ inxptxp . (Contributed by Peter Mazsa, 24-Nov-2018.) $) bropabid $p |- ( x R y <-> ph ) $= ( cv wbr copab cop wcel breqi df-br opabid 3bitri ) BFZCFZDGOPABCHZGOPIQJ AOPDQEKOPQLABCMN $. $} ${ $d A x y $. $d B x y $. $d R x y $. $( Intersection with a Cartesian product. (Contributed by Peter Mazsa, 18-Jul-2019.) $) inxp2 $p |- ( R i^i ( A X. B ) ) = { <. x , y >. | ( ( x e. A /\ y e. B ) /\ x R y ) } $= ( cxp cin cv wbr copab wcel wa wrel wceq relinxp dfrel4v mpbi brinxp2ALTV opabbii eqtri ) ECDFGZAHZBHZUAIZABJZUBCKUCDKLUBUCEILZABJUAMUAUENCDEOABUAP QUDUFABCDUBUCERST $. $} ${ $d A x y $. opabssi.1 $e |- ( ph -> <. x , y >. e. A ) $. $( Lemma for ~ opabf . (Contributed by Peter Mazsa, 21-Oct-2019.) $) opabssi $p |- ( Rel A -> { <. x , y >. | ph } C_ A ) $= ( wrel cv cop wcel copab ssopab2i opabid2 syl5sseq ) DFBGCGHDIZBCJABCJDAN BCEKBCDLM $. $} ${ $d x y $. opabf.1 $e |- -. ph $. $( A class abstraction of a collection of ordered pairs with a negated wff is the empty set. (Contributed by Peter Mazsa, 21-Oct-2019.) $) opabf $p |- { <. x , y >. | ph } = (/) $= ( copab c0 wss wceq wrel rel0 cv cop wcel pm2.21i opabssi ax-mp ss0 ) ABC EZFGZRFHFISJABCFABKCKLFMDNOPRQP $. $} $( The empty-coset of a class is the empty set. (Contributed by Peter Mazsa, 19-May-2019.) $) ec0 $p |- [ A ] (/) = (/) $= ( c0 cec csn cima df-ec 0ima eqtri ) ABCBADZEBABFIGH $. ${ $d R x y $. $( Quotient set with the empty set. (Contributed by Peter Mazsa, 14-Sep-2019.) $) 0qs $p |- ( (/) /. R ) = (/) $= ( vy vx c0 cqs cv cec wceq wrex cab df-qs rex0 abf eqtri ) DAEBFCFAGHZCDI ZBJDCBDAKPBOCLMN $. $} $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Tail Cartesian product (supplement) =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) ${ $d A x $. $d A y $. $d B x $. $d B y $. $d C x $. $d C y $. $d R x $. $d S y $. $d V x $. $d V y $. $d W x $. $d W y $. $d X x $. $d X y $. $( Characterize a trinary relationship over a tail Cartesian product. Together with ~ txpss3v , this completely defines elementhood in a tail cross. (If you need ~ brtxp , use this theorem with ~ el3v or ~ mp3an instead.) (Contributed by Peter Mazsa, 27-Jun-2021.) $) brtxpALTV $p |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A ( R (x) S ) <. B , C >. <-> ( A R B /\ A S C ) ) ) $= ( vx vy wcel wbr c1st cvv wa wb wex mpan2 elv syl5bb w3a cop ctxp cxp cin cres ccnv ccom c2nd df-txp breqi a1i brin wceq opex brcog 3ad2ant1 brcnvg cv opelvvg biantrurd brresALTV syl6rbbr br1steqORIG 3adant1 anbi1d exbidv ancom bitrd breq2 ceqsexgv 3ad2ant2 3bitrd br2ndeqORIG 3ad2ant3 anbi12d ) AFKZBGKZCHKZUAZABCUBZDEUCZLZAWAMNNUDZUFZUGZDUHZUIWDUFZUGZEUHZUEZLZAWAWGLZ AWAWJLZOZABDLZACELZOWCWLPVTAWAWBWKDEUJUKULWLWOPVTAWAWGWJUMULVTWMWPWNWQVTW MAIUSZDLZWRWAWFLZOZIQZWRBUNZWSOZIQZWPVQVRWMXBPZVSVQWANKZXFBCUOZIAWAWFDFNU PRUQVTXAXDIXAWTWSOVTXDWSWTVHVTWTXCWSWTWAWRWELZVTXCWTXIPZIWRNKXGXJXHWRWANN WEURRSVRVSXIXCPVQVRVSOZXIWAWRMLZXCXKXLWAWDKZXLOZXIXKXMXLBCGHUTZVAXIXNPIWD WAWRMNVBSVCBCWRGHVDVIVETVFTVGVRVQXEWPPVSWSWPIBGWRBADVJVKVLVMVTWNAJUSZELZX PWAWILZOZJQZXPCUNZXQOZJQZWQVQVRWNXTPZVSVQXGYDXHJAWAWIEFNUPRUQVTXSYBJXSXRX QOVTYBXQXRVHVTXRYAXQXRWAXPWHLZVTYAXRYEPZJXPNKXGYFXHXPWANNWHURRSVRVSYEYAPV QXKYEWAXPUILZYAXKYGXMYGOZYEXKXMYGXOVAYEYHPJWDWAXPUINVBSVCBCXPGHVNVIVETVFT VGVSVQYCWQPVRXQWQJCHXPCAEVJVKVOVMVPVM $. $} ${ $d A x y $. $d B x y $. $d R x y $. $d S x y $. $d V x y $. $( The binary relationship over a tail cross when the second argument is not an ordered pair. (Contributed by Peter Mazsa, 14-Oct-2020.) $) brtxp2ALTV $p |- ( A e. V -> ( A ( R (x) S ) B <-> E. x E. y ( B = <. x , y >. /\ A R x /\ A S y ) ) ) $= ( ctxp wbr cv cop wceq wa wex wcel w3a cvv cxp txpss3v brel elvv pm4.71ri simprd sylib 19.41vv bitr4i breq2 2exbii bitri wb brtxpALTV el3v23 anbi2d pm5.32i 3anass syl6bbr 2exbidv syl5bb ) CDEFHZIZDAJZBJZKZLZCVCUSIZMZBNANZ CGOZVDCVAEIZCVBFIZPZBNANUTVDUTMZBNANZVGUTVDBNANZUTMVMUTVNUTDQQRZOZVNUTCQO VPCDQVOUSEFSTUCABDUAUDUBVDUTABUEUFVLVFABVDUTVEDVCCUSUGUNUHUIVHVFVKABVHVFV DVIVJMZMVKVHVEVQVDVHVEVQUJABCVAVBEFGQQUKULUMVDVIVJUOUPUQUR $. $} $( Equality theorem for tail Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.) $) txpeq1 $p |- ( A = B -> ( A (x) C ) = ( B (x) C ) ) $= ( wceq c1st cvv cxp cres ccnv ccom c2nd ctxp coeq2 ineq1d df-txp 3eqtr4g cin ) ABDZEFFGZHIZAJZKSHICJZQTBJZUBQACLBCLRUAUCUBABTMNACOBCOP $. ${ txpeq1i.1 $e |- A = B $. $( Equality theorem for tail Cartesian product, inference form. (Contributed by Peter Mazsa, 16-Dec-2020.) $) txpeq1i $p |- ( A (x) C ) = ( B (x) C ) $= ( wceq ctxp txpeq1 ax-mp ) ABEACFBCFEDABCGH $. $} ${ txpeq1d.1 $e |- ( ph -> A = B ) $. $( Equality theorem for tail Cartesian product, deduction form. (Contributed by Peter Mazsa, 7-Sep-2021.) $) txpeq1d $p |- ( ph -> ( A (x) C ) = ( B (x) C ) ) $= ( wceq ctxp txpeq1 syl ) ABCFBDGCDGFEBCDHI $. $} $( Equality theorem for tail Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.) $) txpeq2 $p |- ( A = B -> ( C (x) A ) = ( C (x) B ) ) $= ( wceq c1st cvv cxp cres ccnv ccom c2nd ctxp coeq2 ineq2d df-txp 3eqtr4g cin ) ABDZEFFGZHICJZKSHIZAJZQTUABJZQCALCBLRUBUCTABUAMNCAOCBOP $. ${ txpeq2i.1 $e |- A = B $. $( Equality theorem for tail Cartesian product, inference form. (Contributed by Peter Mazsa, 16-Dec-2020.) $) txpeq2i $p |- ( C (x) A ) = ( C (x) B ) $= ( wceq ctxp txpeq2 ax-mp ) ABECAFCBFEDABCGH $. $} ${ txpeq2d.1 $e |- ( ph -> A = B ) $. $( Equality theorem for tail Cartesian product, deduction form. (Contributed by Peter Mazsa, 7-Sep-2021.) $) txpeq2d $p |- ( ph -> ( C (x) A ) = ( C (x) B ) ) $= ( wceq ctxp txpeq2 syl ) ABCFDBGDCGFEBCDHI $. $} $( Equality theorem for tail Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.) $) txpeq12 $p |- ( ( A = B /\ C = D ) -> ( A (x) C ) = ( B (x) D ) ) $= ( wceq ctxp txpeq1 txpeq2 sylan9eq ) ABECDEACFBCFBDFABCGCDBHI $. ${ txpeq12i.1 $e |- A = B $. txpeq12i.2 $e |- C = D $. $( Equality theorem for tail Cartesian product, inference form. (Contributed by Peter Mazsa, 16-Dec-2020.) $) txpeq12i $p |- ( A (x) C ) = ( B (x) D ) $= ( wceq ctxp txpeq12 mp2an ) ABGCDGACHBDHGEFABCDIJ $. $} ${ txpeq12d.1 $e |- ( ph -> A = B ) $. txpeq12d.2 $e |- ( ph -> C = D ) $. $( Equality theorem for tail Cartesian product, deduction form. (Contributed by Peter Mazsa, 18-Dec-2021.) $) txpeq12d $p |- ( ph -> ( A (x) C ) = ( B (x) D ) ) $= ( wceq ctxp txpeq12 syl2anc ) ABCHDEHBDICEIHFGBCDEJK $. $} ${ $d A x y $. $d B x y $. $d R x y $. $d S x y $. $d V x y $. $( Elementhood in the ` ( R (x) S ) ` -coset of ` A ` . (Contributed by Peter Mazsa, 18-Apr-2020.) (Revised by Peter Mazsa, 21-Sep-2021.) $) electxp $p |- ( A e. V -> ( B e. [ A ] ( R (x) S ) <-> E. x E. y ( B = <. x , y >. /\ A R x /\ A S y ) ) ) $= ( ctxp cec wcel wbr cv cop wceq w3a wex wrel wb txprel relelec brtxp2ALTV ax-mp syl5bb ) DCEFHZIJZCDUDKZCGJDALZBLZMNCUGEKCUHFKOBPAPUDQUEUFREFSDCUDT UBABCDEFGUAUC $. $} ${ $d A x y z $. $d R x y z $. $d S x y z $. $d V x y z $. $( The ` ( R (x) S ) ` -coset of ` A ` . (Contributed by Peter Mazsa, 18-Apr-2020.) (Revised by Peter Mazsa, 21-Sep-2021.) $) ectxp $p |- ( A e. V -> [ A ] ( R (x) S ) = { <. y , z >. | ( A R y /\ A S z ) } ) $= ( vx wcel ctxp cec cv wbr wa copab cop wceq wex w3a electxp 3anass 2exbii syl6bb elopab syl6bbr eqrdv ) CFHZGCDEIJZCAKZDLZCBKZELZMZABNZUFGKZUGHZUNU HUJOPZULMZBQAQZUNUMHUFUOUPUIUKRZBQAQURABCUNDEFSUSUQABUPUIUKTUAUBULABUNUCU DUE $. $} ${ $d A u x y z $. $d B u x y z $. $d C u x y z $. $d R u x y z $. $d S u x y z $. $( Intersection of a tail Cartesian product with a Cartesian product. (Contributed by Peter Mazsa, 7-Apr-2020.) $) txpinxp $p |- ( ( R (x) S ) i^i ( A X. ( B X. C ) ) ) = `' { <. <. y , z >. , u >. | ( ( y e. B /\ z e. C ) /\ ( u e. A /\ u ( R (x) S ) <. y , z >. ) ) } $= ( vx ctxp cxp cin cv wcel wbr wa copab ccnv cop coprab w3a df-3an 3anan12 inxp2 bitr3i opabbii eqtri cnvopab breq2 anbi2d dfoprab4 cnveqi 3eqtr2i wceq ) GHJZDEFKZKLZIMZUPNZCMZDNZUTURUOOZPZPZCIQZVDICQZRAMZENBMZFNPVAUTVGV HSZUOOZPZPABCTZRUQVAUSPVBPZCIQVECIDUPUOUDVMVDCIVMVAUSVBUAVDVAUSVBUBVAUSVB UCUEUFUGVDICUHVFVLVCVKABCIEFURVIUNVBVJVAURVIUTUOUIUJUKULUM $. $} ${ $d A u x $. $d B u x $. $d C u x $. $d R u x $. $d S u x $. $( Intersection of a tail Cartesian product with a Cartesian product. (Contributed by Peter Mazsa, 8-Apr-2020.) $) txpinxp2 $p |- ( ( R (x) S ) i^i ( A X. ( B X. C ) ) ) = { <. u , x >. | ( x e. ( B X. C ) /\ ( u e. A /\ u ( R (x) S ) x ) ) } $= ( ctxp cxp cin cv wcel wa wbr copab inxp2 w3a 3ancoma df-3an 3anass eqtri 3bitr3i opabbii ) FGHZCDEIZIJBKZCLZAKZUELZMUFUHUDNZMZBAOUIUGUJMMZBAOBACUE UDPUKULBAUGUIUJQUIUGUJQUKULUGUIUJRUGUIUJSUIUGUJTUBUCUA $. $} $( Sethood condition for the intersection of a tail Cartesian product with a Cartesian product. (Contributed by Peter Mazsa, 12-Apr-2020.) $) txpinxpex $p |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( R (x) S ) i^i ( A X. ( B X. C ) ) ) e. _V ) $= ( wcel w3a cxp cvv wa ctxp cin 3anass xpexg anim2i sylbi inxpex olcs syl ) AFIZBGIZCHIZJZUCBCKZLIZMZDENZAUGKOLIZUFUCUDUEMZMUIUCUDUEPULUHUCBCGHQRSUJLIU IUKAUGUJFLLTUAUB $. ${ $d A u x y z $. $d B u x y z $. $d C u x y z $. $d R u x y z $. $d S u x y z $. $( Two ways to express the intersection of a tail Cartesian product with a Cartesian product. (Contributed by Peter Mazsa, 10-Apr-2020.) $) inxptxp $p |- ( ( R i^i ( A X. B ) ) (x) ( S i^i ( A X. C ) ) ) = ( ( R (x) S ) i^i ( A X. ( B X. C ) ) ) $= ( vu vx vy vz cxp cin ctxp cv wbr wb wcel wa wex anbi2i 3bitri brtxp2ALTV wceq wal cop w3a cvv elv txpinxp2 bropabid 3anass 2exbii brinxp2ALTV anan anbi12i bitri anass eqelb opelxp bitr2i anbi1i 3bitr2i ancom an12 19.42vv bitr4i 3bitr4ri gen2 wrel txprel relinxp eqrel2 mp2an mpbir ) DABJKZEACJK ZLZDELZABCJZJKZUBZFMZGMZVPNZWAWBVSNZOZGUCFUCZWEFGWBVRPZWAAPZWAWBVQNZQZQZW GWHWBHMZIMZUDZUBZWAWLDNZWAWMENZUEZIRHRZQZQZWDWCWJWTWGWIWSWHWIWSOFHIWAWBDE UFUAUGSSWKFGVSGFABCDEUHUIWCWOWAWLVNNZWAWMVONZUEZIRHRZWOXBXCQZQZIRHRZXAWCX EOFHIWAWBVNVOUFUAUGXDXGHIWOXBXCUJUKXHWGWHWRQZQZIRHRWGXIIRHRZQXAXGXJHIXGWG WOWHWPWQQZQZQZQZXJXGWOWGQZXMQZWGWOQZXMQXOXGWOWLBPZWMCPZQZXMQZQWOYAQZXMQXQ XFYBWOXFWHXSQWPQZWHXTQWQQZQYBXBYDXCYEABWAWLDULACWAWMEULUNWHXSWPXTWQUMUOSW OYAXMUPYCXPXMXPWOWNVRPZQYCWBWNVRUQYFYAWOWLWMBCURSUSUTVAXPXRXMWOWGVBUTWGWO XMUPTXNXIWGXNWHWOXLQZQXIWOWHXLVCWRYGWHWOWPWQUJSVESUOUKWGXIHIVDXKWTWGWHWRH IVDSTTVFVGVPVHVSVHVTWFOVNVOVIAVRVQVJFGVPVSVKVLVM $. $} ${ $d A y z $. $d B y z $. $d R y z $. $d S y z $. $d V y z $. $( The converse of a binary relationship over a tail cross. (Contributed by Peter Mazsa, 11-Jul-2021.) $) br1cnvtxp2 $p |- ( B e. V -> ( A `' ( R (x) S ) B <-> E. y E. z ( A = <. y , z >. /\ B R y /\ B S z ) ) ) $= ( ctxp ccnv wbr wcel cv cop wceq w3a wex wrel wb txprel relbrcnvg syl5bb ax-mp brtxp2ALTV ) CDEFHZIJZDCUDJZDGKCALZBLZMNDUGEJDUHFJOBPAPUDQUEUFREFSC DUDTUBABDCEFGUCUA $. $} ${ $d A y z $. $d B y z $. $d R y z $. $d S y z $. $d V y z $. $( Elementhood in the converse tail Cartesian product coset of ` A ` . (Contributed by Peter Mazsa, 11-Jul-2021.) $) elec1cnvtxp2 $p |- ( B e. V -> ( B e. [ A ] `' ( R (x) S ) <-> E. y E. z ( A = <. y , z >. /\ B R y /\ B S z ) ) ) $= ( ctxp ccnv cec wcel wbr cv cop wceq w3a wex wrel wb relcnv relelec ax-mp br1cnvtxp2 syl5bb ) DCEFHZIZJKZCDUFLZDGKCAMZBMZNODUIELDUJFLPBQAQUFRUGUHSU ETDCUFUAUBABCDEFGUCUD $. $} ${ $d R u w x y $. $d S u w x y $. $( Range of the tail cross of classes. (Contributed by Peter Mazsa, 1-Jun-2020.) $) rntxp $p |- ran ( R (x) S ) = { <. x , y >. | E. u ( u R x /\ u S y ) } $= ( vw cv cop wceq wbr w3a wex cab wa ctxp crn copab 3anass 3exbii abbii c0 exrot3 19.42v 2exbii 3bitri ccnv cec wne dfrn6 n0 wb cvv elec1cnvtxp2 elv wcel exbii bitri eqtri df-opab 3eqtr4i ) FGZAGZBGZHIZCGZVBDJZVEVCEJZKZBLA LZCLZFMZVDVFVGNZCLZNZBLALZFMDEOZPZVMABQVJVOFVJVDVLNZBLALCLVRCLZBLALVOVHVR CABVDVFVGRSVRCABUBVSVNABVDVLCUCUDUETVQVAVPUFUGZUAUHZFMVKFVPUIWAVJFWAVEVTU OZCLVJCVTUJWBVICWBVIUKCABVAVEDEULUMUNUPUQTURVMABFUSUT $. $} ${ $d A u x y $. $d R u x y $. $d S u w x y $. $( Range of the restricted tail cross of classes. (Contributed by Peter Mazsa, 5-Dec-2021.) $) rntxpres $p |- ran ( R (x) ( S |` A ) ) = { <. x , y >. | E. u e. A ( u R x /\ u S y ) } $= ( cres ctxp crn cv wbr wa wex copab wrex rntxp wcel wb cvv bitr4i opabbii brresALTV elv anbi2i an12 exbii df-rex eqtri ) EFDGZHICJZAJEKZUJBJZUIKZLZ CMZABNUKUJULFKZLZCDOZABNABCEUIPUOURABUOUJDQZUQLZCMURUNUTCUNUKUSUPLZLUTUMV AUKUMVARBDUJULFSUBUCUDUSUKUPUETUFUQCDUGTUAUH $. $} ${ $d A u x y $. $d R u x y $. $( Range of the restricted converse epsilon tail cross of classes. (Contributed by Peter Mazsa, 6-Dec-2021.) $) rntxpcnvepres $p |- ran ( R (x) ( `' _E |` A ) ) = { <. x , y >. | E. u e. A ( y e. u /\ u R x ) } $= ( cep ccnv cres ctxp crn cv wbr wa wrex copab wcel rntxpres cvv brcnvep wb elv anbi2ri rexbii opabbii eqtri ) EFGZDHIJCKZAKELZUGBKZUFLZMZCDNZABOU IUGPZUHMZCDNZABOABCDEUFQULUOABUKUNCDUJUMUHUJUMTCUGUIRSUAUBUCUDUE $. $} ${ $d A u x y $. $d R u x y $. $( Range of the restricted identity tail cross of classes. (Contributed by Peter Mazsa, 6-Dec-2021.) $) rntxpidres $p |- ran ( R (x) ( _I |` A ) ) = { <. x , y >. | E. u e. A ( u = y /\ u R x ) } $= ( cid cres ctxp crn cv wbr wa wrex copab wceq rntxpres wb cvv ideqg elv anbi2ri rexbii opabbii eqtri ) EFDGHICJZAJEKZUEBJZFKZLZCDMZABNUEUGOZUFLZC DMZABNABCDEFPUJUMABUIULCDUHUKUFUHUKQBUEUGRSTUAUBUCUD $. $} $( Two ways to express restriction of tail Cartesian product, cf. ~ txpres2 , ~ txpres3 . (Contributed by Peter Mazsa, 5-Jun-2021.) $) txpres $p |- ( ( R (x) S ) |` A ) = ( ( R |` A ) (x) S ) $= ( c1st cvv cxp cres ccnv ccom c2nd ctxp ineq1i df-txp reseq1i inres2 eqtr4i cin resco 3eqtr4i ) DEEFZGHZBIZAGZJTGHCIZQZUABAGZIZUDQBCKZAGZUFCKUCUGUDUABA RLUIUBUDQZAGUEUHUJABCMNAUBUDOPUFCMS $. $( Two ways to express restriction of tail Cartesian product, cf. ~ txpres , ~ txpres3 . (Contributed by Peter Mazsa, 6-Sep-2021.) $) txpres2 $p |- ( ( R (x) S ) |` A ) = ( R (x) ( S |` A ) ) $= ( c1st cvv cxp cres ccnv ccom c2nd resco ineq2i df-txp reseq1i inres eqtr4i cin ctxp 3eqtr4i ) DEEFZGHBIZJTGHZCIZAGZQZUAUBCAGZIZQBCRZAGZBUFRUDUGUAUBCAK LUIUAUCQZAGUEUHUJABCMNUAUCAOPBUFMS $. $( Two ways to express restriction of tail Cartesian product, cf. ~ txpres , ~ txpres2 . (Contributed by Peter Mazsa, 28-Mar-2020.) $) txpres3 $p |- ( ( R (x) S ) |` A ) = ( ( R |` A ) (x) ( S |` A ) ) $= ( c1st cvv cxp cres ccnv ccom c2nd cin ctxp ineq12i df-txp reseq1i resindir resco eqtri 3eqtr4i ) DEEFZGHZBIZAGZJTGHZCIZAGZKZUABAGZIZUDCAGZIZKBCLZAGZUH UJLUCUIUFUKUABAQUDCAQMUMUBUEKZAGUGULUNABCNOUBUEAPRUHUJNS $. $( Two ways to express restriction of tail Cartesian product. (Contributed by Peter Mazsa, 29-Dec-2020.) $) txpres4 $p |- ( ( R (x) S ) |` A ) = ( ( R (x) S ) i^i ( A X. ( ran ( R |` A ) X. ran ( S |` A ) ) ) ) $= ( ctxp cres crn cxp cin txpres3 dfres4 txpeq12i inxptxp 3eqtri ) BCDZAEBAEZ CAEZDBAOFZGHZCAPFZGHZDNAQSGGHABCIORPTABJACJKAQSBCLM $. $( Sethood condition for the restricted tail cross of classes. (Contributed by Peter Mazsa, 16-Dec-2020.) (Revised by Peter Mazsa, 7-Sep-2021.) $) txpresex $p |- ( ( A e. V /\ R e. W /\ ( S |` A ) e. X ) -> ( R (x) ( S |` A ) ) e. _V ) $= ( wcel cres w3a ctxp cvv crn cxp cin simp1 resexg rnexg syl dfres4 eleq1i 3ad2ant2 3ad2ant3 inxptxp txpinxpex syl5eqel txpeq12i sylibr txpres3 eqtr3i 3jca txpres2 sylib ) ADGZBEGZCAHZFGZIZBAHZUOJZKGZBUOJZKGUQBAURLZMNZCAUOLZMN ZJZKGZUTUQUMVBKGZVDKGZIZVGUQUMVHVIUMUNUPOUNUMVHUPUNURKGVHBAEPURKQRUAUPUMVIU NUOFQUBUJVJVFBCJZAVBVDMMNKAVBVDBCUCAVBVDBCDKKUDUERUSVFKURVCUOVEABSACSUFTUGU SVAKVKAHUSVAABCUHABCUKUITUL $. $( Sethood condition for the tail cross restricted identity. (Contributed by Peter Mazsa, 31-Dec-2021.) $) txpidresex $p |- ( ( A e. V /\ R e. W ) -> ( R (x) ( _I |` A ) ) e. _V ) $= ( wcel wa cid cres cvv w3a ctxp resiexg adantr ancli df-3an sylibr txpresex syl ) ACEZBDEZFZSTGAHZIEZJZBUBKIEUAUAUCFUDUAUCSUCTACLMNSTUCOPABGCDIQR $. $( Sethood condition for the tail cross restricted converse epsilon. (Contributed by Peter Mazsa, 16-Dec-2020.) (Revised by Peter Mazsa, 23-Sep-2021.) $) txpcnvepresex $p |- ( ( A e. V /\ R e. W ) -> ( R (x) ( `' _E |` A ) ) e. _V ) $= ( wcel cep ccnv cres cvv w3a cnvepresex adantr ancli df-3an sylibr txpresex wa ctxp syl ) ACEZBDEZQZTUAFGZAHZIEZJZBUDRIEUBUBUEQUFUBUETUEUAACKLMTUAUENOA BUCCDIPS $. $( Binary relation on an intersection is a special case of binary relation on tail Cartesian product. (Contributed by Peter Mazsa, 21-Aug-2021.) $) brin2 $p |- ( ( A e. V /\ B e. W ) -> ( A ( R i^i S ) B <-> A ( R (x) S ) <. B , B >. ) ) $= ( wcel wa cop ctxp wbr cin wb brtxpALTV 3anidm23 brin syl6rbbr ) AEGZBFGZHA BBICDJKZABCKABDKHZABCDLKRSTUAMABBCDEFFNOABCDPQ $. $( Binary relation on an intersection is a special case of binary relation on tail Cartesian product. (Contributed by Peter Mazsa, 21-Aug-2021.) $) brin3 $p |- ( ( A e. V /\ B e. W ) -> ( A ( R i^i S ) B <-> A ( R (x) S ) { { B } } ) ) $= ( wcel wa cin wbr cop ctxp csn brin2 wceq opidORIG adantl breq2d bitrd ) AE GZBFGZHZABCDIJABBKZCDLZJABMMZUDJABCDEFNUBUCUEAUDUAUCUEOTBFPQRS $. $( Lemma for ~ trcoss . (Contributed by Peter Mazsa, 2-Oct-2018.) $) motr $p |- ( E* x ps -> ( ( E. x ( ph /\ ps ) /\ E. x ( ps /\ ch ) ) -> E. x ( ph /\ ch ) ) ) $= ( wmo wa wex wi w3a exancom anbi1i anbi2i 3anass bitr4i mopick2 sylbi exbii exsimpr syl impexp mpbi ) BDEZABFDGZBCFDGZFZFZACFZDGZHUBUEUHHHUFBACIZDGZUHU FUBBAFDGZUDIZUJUFUBUKUDFZFULUEUMUBUCUKUDABDJKLUBUKUDMNBACDOPUJBUGFZDGUHUIUN DBACMQBUGDRPSUBUEUHTUA $. $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Cosets by ` R ` =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) ${ $d R u x y $. $( Define the class of cosets by ` R ` : ` x ` and ` y ` are cosets by ` R ` iff there exists a set ` u ` such that both ` u R x ` and ` u R y ` hold, i.e., both ` x ` and ` y ` are are elements of the ` R ` -coset of ` u ` (cf. ~ dfcoss2 and the comment of ~ dfec2 ). ` R ` is usually a relation. This concept simplifies theorems relating partition and equivalence: the left side of these theorems relate to ` R ` , the right side relate to ` ,~ R ` (cf. e.g. ~ pet ). Without the definition of ` ,~ R ` we should have to relate the right side of these theorems to a composition of a converse (cf. ~ dfcoss3 ) or to the range of a tail cross of classes (cf. ~ dfcoss4 ), which would make the theorems complicated and confusing. Alternate definition is ~ dfcoss2 . Technically, we can define it via composition ( ~ dfcoss3 ) or as the range of a tail cross ( ~ dfcoss4 ), but neither of these definitions reveal directly how the cosets by ` R ` relate to each other. We define functions ( ~ df-funsALTV , ~ df-funALTV ) and disjoints ( ~ dfdisjs , ~ dfdisjs2 , ~ df-disjALTV , ~ dfdisjALTV2 ) with the help of it as well. (Contributed by Peter Mazsa, 9-Jan-2018.) $) df-coss $a |- ,~ R = { <. x , y >. | E. u ( u R x /\ u R y ) } $. $} $( Define the class of coelements on the class ` A ` , cf. the alternate definition ~ dfcoels . Possible definitions are the special cases of ~ dfcoss3 and ~ dfcoss4 . (Contributed by Peter Mazsa, 20-Nov-2019.) $) df-coels $a |- ~ A = ,~ ( `' _E |` A ) $. ${ $d R u x y $. $( Alternate definition of the class of cosets by ` R ` : ` x ` and ` y ` are cosets by ` R ` iff there exists a set ` u ` such that both ` x ` and ` y ` are are elements of the ` R ` -coset of ` u ` (cf. the comment of ~ dfec2 ). ` R ` is usually a relation. (Contributed by Peter Mazsa, 16-Jan-2018.) $) dfcoss2 $p |- ,~ R = { <. x , y >. | E. u ( x e. [ u ] R /\ y e. [ u ] R ) } $= ( ccoss cv wbr wa wex copab cec wcel df-coss wb cvv elecALTV el2v anbi12i exbii opabbii eqtr4i ) DECFZAFZDGZUBBFZDGZHZCIZABJUCUBDKZLZUEUILZHZCIZABJ ABCDMUMUHABULUGCUJUDUKUFUJUDNCAUBUCDOOPQUKUFNCBUBUEDOOPQRSTUA $. $} ${ $d R u x y $. $( A theorem which technically is an alternate definition of the class of cosets by ` R ` (cf. the comment of ~ df-coss ). (Contributed by Peter Mazsa, 27-Dec-2018.) $) dfcoss3 $p |- ,~ R = ( R o. `' R ) $= ( vx vu vy cv ccnv wbr wa wex copab ccom ccoss wb cvv brcnvg anbi1i exbii el2v opabbii df-co df-coss 3eqtr4ri ) BEZCEZAFZGZUDDEAGZHZCIZBDJUDUCAGZUG HZCIZBDJAUEKALUIULBDUHUKCUFUJUGUFUJMBCUCUDNNAORPQSBDCAUETBDCAUAUB $. $} ${ $d R u x y $. $( A theorem which technically is an alternate definition of the class of cosets by ` R ` (cf. the comment of ~ df-coss ). (Contributed by Peter Mazsa, 12-Jul-2021.) $) dfcoss4 $p |- ,~ R = ran ( R (x) R ) $= ( vu vx vy ccoss cv wbr wa wex copab ctxp crn df-coss rntxp eqtr4i ) AEBF ZCFAGPDFAGHBICDJAAKLCDBAMCDBAANO $. $} $( If ` A ` is a set then the class of cosets by ` A ` is a set. (Contributed by Peter Mazsa, 4-Jan-2019.) $) cossex $p |- ( A e. V -> ,~ A e. _V ) $= ( wcel ccoss ccnv ccom cvv dfcoss3 cnvexg coexg mpdan syl5eqel ) ABCZADAAEZ FZGAHMNGCOGCABIANBGJKL $. $( If ` A ` is a set then the class of cosets by the converse of ` A ` is a set. (Contributed by Peter Mazsa, 18-Oct-2019.) $) cosscnvex $p |- ( A e. V -> ,~ `' A e. _V ) $= ( wcel ccnv cvv ccoss cnvexg cossex syl ) ABCADZECJFECABGJEHI $. $( Sethood condition for restricted converse epsilon cosets. (Contributed by Peter Mazsa, 24-Sep-2021.) $) 1cosscnvepresex $p |- ( A e. V -> ,~ ( `' _E |` A ) e. _V ) $= ( wcel cep ccnv cres cvv ccoss cnvepresex cossex syl ) ABCDEAFZGCLHGCABILGJ K $. $( Sethood condition for restricted converse epsilon tail cross cosets. (Contributed by Peter Mazsa, 23-Sep-2021.) $) 1cosstxpcnvepresex $p |- ( ( A e. V /\ R e. W ) -> ,~ ( R (x) ( `' _E |` A ) ) e. _V ) $= ( wcel wa cep ccnv cres ctxp cvv ccoss txpcnvepresex cossex syl ) ACEBDEFBG HAIJZKEPLKEABCDMPKNO $. ${ $d R u x y $. $( Cosets by ` R ` is a relation. (Contributed by Peter Mazsa, 27-Dec-2018.) $) relcoss $p |- Rel ,~ R $= ( vu vx vy cv wbr wa wex ccoss df-coss relopabi ) BEZCEAFLDEAFGBHCDAICDBA JK $. $} $( Coelements on ` A ` is a relation. (Contributed by Peter Mazsa, 5-Oct-2021.) $) relcoels $p |- Rel ~ A $= ( ccoels wrel cep ccnv cres ccoss relcoss df-coels releqi mpbir ) ABZCDEAFZ GZCMHLNAIJK $. ${ $d A u x y $. $d B u x y $. $( Subclass theorem for the classes of cosets by ` A ` and ` B ` . (Contributed by Peter Mazsa, 11-Nov-2019.) $) cossss $p |- ( A C_ B -> ,~ A C_ ,~ B ) $= ( vu vx vy wss cv wbr wa wex copab ccoss anim12d eximdv ssopab2dv df-coss brss 3sstr4g ) ABFZCGZDGZAHZTEGZAHZIZCJZDEKTUABHZTUCBHZIZCJZDEKALBLSUFUJD ESUEUICSUBUGUDUHTUAABQTUCABQMNODECAPDECBPR $. $} ${ $d A u x y $. $d B u x y $. $( Equality theorem for the classes of cosets by ` A ` and ` B ` . (Contributed by Peter Mazsa, 9-Jan-2018.) $) cosseq $p |- ( A = B -> ,~ A = ,~ B ) $= ( vu vx vy wceq cv wbr wa wex copab ccoss anbi12d exbidv opabbidv df-coss breq 3eqtr4g ) ABFZCGZDGZAHZTEGZAHZIZCJZDEKTUABHZTUCBHZIZCJZDEKALBLSUFUJD ESUEUICSUBUGUDUHTUAABQTUCABQMNODECAPDECBPR $. $} ${ cosseqi.1 $e |- A = B $. $( Equality theorem for the classes of cosets by ` A ` and ` B ` , inference form. (Contributed by Peter Mazsa, 9-Jan-2018.) $) cosseqi $p |- ,~ A = ,~ B $= ( wceq ccoss cosseq ax-mp ) ABDAEBEDCABFG $. $} ${ cosseqd.1 $e |- ( ph -> A = B ) $. $( Equality theorem for the classes of cosets by ` A ` and ` B ` , deduction form. (Contributed by Peter Mazsa, 4-Nov-2019.) $) cosseqd $p |- ( ph -> ,~ A = ,~ B ) $= ( wceq ccoss cosseq syl ) ABCEBFCFEDBCGH $. $} ${ $d A u x y $. $d R u x y $. $( The class of cosets by a restriction. (Contributed by Peter Mazsa, 20-Apr-2019.) $) 1cossres $p |- ,~ ( R |` A ) = { <. x , y >. | E. u e. A ( u R x /\ u R y ) } $= ( cres ccoss cv wbr wa wex copab wrex df-coss df-rex wb cvv brresALTV elv wcel anandi anbi12i bitr4i exbii bitri opabbii eqtr4i ) EDFZGCHZAHZUHIZUI BHZUHIZJZCKZABLUIUJEIZUIULEIZJZCDMZABLABCUHNUSUOABUSUIDTZURJZCKUOURCDOVAU NCVAUTUPJZUTUQJZJUNUTUPUQUAUKVBUMVCUKVBPADUIUJEQRSUMVCPBDUIULEQRSUBUCUDUE UFUG $. $} ${ $d A u x y $. $( Alternate definition of the class of coelements on the class ` A ` . (Contributed by Peter Mazsa, 20-Apr-2019.) $) dfcoels $p |- ~ A = { <. x , y >. | E. u e. A ( x e. u /\ y e. u ) } $= ( ccoels cep ccnv cres ccoss cv wcel wa wrex copab wbr wb cvv brcnvep elv eqtri df-coels 1cossres anbi12i rexbii opabbii ) DEFGZDHIZAJZCJZKZBJZUIKZ LZCDMZABNZDUAUGUIUHUFOZUIUKUFOZLZCDMZABNUOABCDUFUBUSUNABURUMCDUPUJUQULUPU JPCUIUHQRSUQULPCUIUKQRSUCUDUETT $. $} ${ $d A u x y $. $d B u x y $. $d R u x y $. $d V u $. $d W u $. $( ` A ` and ` B ` are cosets by ` R ` : a binary relation. (Contributed by Peter Mazsa, 27-Dec-2018.) $) brcoss $p |- ( ( A e. V /\ B e. W ) -> ( A ,~ R B <-> E. u ( u R A /\ u R B ) ) ) $= ( vx vy cv wbr wa wex ccoss wceq breq2 bi2anan9 exbidv df-coss brabga ) A IZGIZDJZTHIZDJZKZALTBDJZTCDJZKZALGHBCDMEFUABNZUCCNZKUEUHAUIUBUFUJUDUGUABT DOUCCTDOPQGHADRS $. $} ${ $d A u $. $d B u $. $d R u $. $d V u $. $d W u $. $( Alternate form of the ` A ` and ` B ` are cosets by ` R ` binary relation. (Contributed by Peter Mazsa, 26-Mar-2019.) $) brcoss2 $p |- ( ( A e. V /\ B e. W ) -> ( A ,~ R B <-> E. u ( A e. [ u ] R /\ B e. [ u ] R ) ) ) $= ( wcel wa ccoss wbr cv wex cec brcoss exan3 bitr4d ) BEGCFGHBCDIJAKZBDJQC DJHALBQDMZGCRGHALABCDEFNABCDEFOP $. $} ${ $d A u $. $d B u $. $d R u $. $d V u $. $d W u $. $( Alternate form of the ` A ` and ` B ` are cosets by ` R ` binary relation. (Contributed by Peter Mazsa, 26-Mar-2019.) $) brcoss3 $p |- ( ( A e. V /\ B e. W ) -> ( A ,~ R B <-> ( [ A ] `' R i^i [ B ] `' R ) =/= (/) ) ) $= ( vu wcel wa cv ccnv wbr wex cec cin c0 wne wb cvv brcnvg el2v2 bi2anan9 ccoss exbidv ecinn0 brcoss 3bitr4rd ) ADGZBEGZHZAFIZCJZKZBUJUKKZHZFLUJACK ZUJBCKZHZFLAUKMBUKMNOPABCUBKUIUNUQFUGULUOUHUMUPUGULUOQFAUJDRCSTUHUMUPQFBU JERCSTUAUCFABUKDEUDFABCDEUEUF $. $} ${ $d A u $. $d B u $. $d R u $. $d V u $. $d W u $. $( For sets, the ` A ` and ` B ` cosets by ` R ` binary relation and the ` B ` and ` A ` cosets by ` R ` binary relation are the same. (Contributed by Peter Mazsa, 27-Dec-2018.) $) brcosscnvcoss $p |- ( ( A e. V /\ B e. W ) -> ( A ,~ R B <-> B ,~ R A ) ) $= ( vu wcel wa cv wbr wex ccoss wb exancom a1i brcoss ancoms 3bitr4d ) ADGZ BEGZHZFIZACJZUBBCJZHFKZUDUCHFKZABCLZJBAUGJZUEUFMUAUCUDFNOFABCDEPTSUHUFMFB ACEDPQR $. $} ${ $d A u x y $. $d B u x y $. $d C u x y $. $( ` B ` and ` C ` are coelements : a binary relation. (Contributed by Peter Mazsa, 14-Jan-2020.) (Revised by Peter Mazsa, 5-Oct-2021.) $) brcoels $p |- ( ( B e. V /\ C e. W ) -> ( B ~ A C <-> E. u e. A ( B e. u /\ C e. u ) ) ) $= ( vx vy cv wcel wa wrex ccoels wceq eleq1 bi2anan9 rexbidv dfcoels brabga ) GIZAIZJZHIZUAJZKZABLCUAJZDUAJZKZABLGHCDBMEFTCNZUCDNZKUEUHABUIUBUFUJUDUG TCUAOUCDUAOPQGHABRS $. $} ${ $d R x y z $. $d S x y z $. $( Two ways of saying that cosets by cosets by ` R ` is a subclass. (Contributed by Peter Mazsa, 17-Sep-2021.) $) cocossss $p |- ( ,~ ,~ R C_ S <-> A. x A. y A. z ( ( x ,~ R y /\ y ,~ R z ) -> x S z ) ) $= ( ccoss wss cv wbr wi wal wa wrel wb relcoss wex cvv el2v bitri albii ssrel3 ax-mp brcoss brcosscnvcoss anbi1i exbii imbi1i 19.23v bitr4i alcom ) DFZFZEGZAHZCHZULIZUNUOEIZJZCKZAKZUNBHZUKIZVAUOUKIZLZUQJZCKBKZAKULMUMUTN UKOACULEUAUBUSVFAUSVEBKZCKVFURVGCURVDBPZUQJVGUPVHUQUPVAUNUKIZVCLZBPZVHUPV KNACBUNUOUKQQUCRVJVDBVIVBVCVIVBNBAVAUNDQQUDRUEUFSUGVDUQBUHUITVECBUJSTS $. $} ${ $d R x y $. $( The converse of cosets by ` R ` are cosets by ` R ` . (Contributed by Peter Mazsa, 3-May-2019.) $) cnvcosseq $p |- `' ,~ R = ,~ R $= ( vx vy ccoss ccnv wss wceq cv wbr wal cvv brcosscnvcoss el2v biimpi gen2 wi wb cnvsym mpbir wrel relcoss relcnveq ax-mp mpbi ) ADZEZUEFZUFUEGZUGBH ZCHZUEIZUJUIUEIZPZCJBJUMBCUKULUKULQBCUIUJAKKLMNOBCUERSUETUGUHQAUAUEUBUCUD $. $} $( Cosets by ` ,~ R ` binary relation. (Contributed by Peter Mazsa, 25-Aug-2019.) $) br2coss $p |- ( ( A e. V /\ B e. W ) -> ( A ,~ ,~ R B <-> ( [ A ] ,~ R i^i [ B ] ,~ R ) =/= (/) ) ) $= ( wcel wa ccoss wbr ccnv cec cin c0 brcoss3 cnvcosseq eceq2i ineq12i neeq1i wne syl6bb ) ADFBEFGABCHZHIAUAJZKZBUBKZLZMSAUAKZBUAKZLZMSABUADENUEUHMUCUFUD UGUBUAACOZPUBUABUIPQRT $. ${ $d A u $. $d B u $. $d C u $. $d R u $. $d V u $. $d W u $. $( ` B ` and ` C ` are cosets by restriction: a binary relation. (Contributed by Peter Mazsa, 30-Dec-2018.) $) br1cossres $p |- ( ( B e. V /\ C e. W ) -> ( B ,~ ( R |` A ) C <-> E. u e. A ( u R B /\ u R C ) ) ) $= ( wcel wa cres ccoss wbr cv wex wrex brcoss exanres bitrd ) CFHDGHICDEBJZ KLAMZCSLTDSLIANTCELTDELIABOACDSFGPABCDEEFGQR $. $} ${ $d A x $. $d B x $. $d C x $. $d R x $. $d V x $. $d W x $. $( ` B ` and ` C ` are cosets by restriction: a binary relation. (Contributed by Peter Mazsa, 3-Jan-2018.) $) br1cossres2 $p |- ( ( B e. V /\ C e. W ) -> ( B ,~ ( R |` A ) C <-> E. x e. A ( B e. [ x ] R /\ C e. [ x ] R ) ) ) $= ( wcel wa cres ccoss wbr cv wrex cec br1cossres exanres3 bitr4d ) CFHDGHI CDEBJKLAMZCELSDELIABNCSEOZHDTHIABNABCDEFGPABCDEEFGQR $. $} ${ $d A x $. $d B x $. $d R x $. $d V x $. $d W x $. $( ` A ` and ` B ` are cosets by relation ` R ` : a binary relation. (Contributed by Peter Mazsa, 22-Apr-2021.) $) relbrcoss $p |- ( ( A e. V /\ B e. W ) -> ( Rel R -> ( A ,~ R B <-> E. x e. dom R ( A e. [ x ] R /\ B e. [ x ] R ) ) ) ) $= ( wcel wa wrel ccoss wbr cv cec cdm wrex wb cres resdm cosseqd breqd ex adantl br1cossres2 adantr bitr3d ) BEGCFGHZDIZBCDJZKZBALDMZGCUJGHADNZOZPU FUGHBCDUKQZJZKZUIULUGUOUIPUFUGUNUHBCUGUMDDRSTUBUFUOULPUGAUKBCDEFUCUDUEUA $. $} ${ $d A u $. $d B u $. $d C u $. $d R u $. $d S u $. $d V u $. $d W u $. $( ` B ` and ` C ` are cosets by intersection with restriction: a binary relation. (Contributed by Peter Mazsa, 31-Dec-2021.) $) br1cossinres $p |- ( ( B e. V /\ C e. W ) -> ( B ,~ ( R i^i ( S |` A ) ) C <-> E. u e. A ( ( u S B /\ u R B ) /\ ( u S C /\ u R C ) ) ) ) $= ( cres cin ccoss wbr wcel wa cv wrex inres cosseqi breqi brin br1cossres anbi12i an2anr bitri rexbii syl6bb syl5bb ) CDEFBIJZKZLCDEFJZBIZKZLZCGMDH MNZAOZCFLZUOCELZNUODFLZUODELZNNZABPZCDUIULUHUKEFBQRSUNUMUOCUJLZUODUJLZNZA BPVAABCDUJGHUAVDUTABVDUQUPNZUSURNZNUTVBVEVCVFUOCEFTUODEFTUBUQUPUSURUCUDUE UFUG $. $} ${ $d A u $. $d B u $. $d C u $. $d D u $. $d E u $. $d R u $. $d S u $. $d V u $. $d W u $. $d X u $. $d Y u $. $( ` <. B , C >. ` and ` <. D , E >. ` are cosets by intersection with restriction: a binary relation. (Contributed by Peter Mazsa, 8-Jun-2021.) $) br1cosstxpres $p |- ( ( ( B e. V /\ C e. W ) /\ ( D e. X /\ E e. Y ) ) -> ( <. B , C >. ,~ ( R (x) ( S |` A ) ) <. D , E >. <-> E. u e. A ( ( u S C /\ u R B ) /\ ( u S E /\ u R D ) ) ) ) $= ( cop cres ctxp wbr wcel wa cvv wb ccoss cv wrex txpres2 breqi br1cossres cosseqi opex mp2an brtxpALTV el3v1 bi2anan9 an2anr syl6bb rexbidv syl5bbr syl5bb ) CDMZEHMZFGBNOZUAZPURUSFGOZBNZUAZPZCIQZDJQZRZEKQZHLQZRZRZAUBZDGPZ VMCFPZRVMHGPZVMEFPZRRZABUCZURUSVDVAVCUTBFGUDUGUEVEVMURVBPZVMUSVBPZRZABUCZ VLVSURSQUSSQVEWCTCDUHEHUHABURUSVBSSUFUIVLWBVRABVLWBVOVNRZVQVPRZRVRVHVTWDV KWAWEVFVGVTWDTAVMCDFGSIJUJUKVIVJWAWETAVMEHFGSKLUJUKULVOVNVQVPUMUNUOUQUP $. $} ${ $d A u $. $d B u $. $d C u $. $d R u $. $d V u $. $d W u $. $( ` B ` and ` C ` are cosets by intersection with the restricted identity class: a binary relation. (Contributed by Peter Mazsa, 31-Dec-2021.) $) br1cossinidres $p |- ( ( B e. V /\ C e. W ) -> ( B ,~ ( R i^i ( _I |` A ) ) C <-> E. u e. A ( ( u = B /\ u R B ) /\ ( u = C /\ u R C ) ) ) ) $= ( wcel wa cid cres cin wbr wrex wceq wb cvv ideq2 elv anbi1i br1cossinres ccoss cv anbi12i rexbii syl6bb ) CFHDGHICDEJBKLUBMAUCZCJMZUGCEMZIZUGDJMZU GDEMZIZIZABNUGCOZUIIZUGDOZULIZIZABNABCDEJFGUAUNUSABUJUPUMURUHUOUIUHUOPAUG CQRSTUKUQULUKUQPAUGDQRSTUDUEUF $. $} ${ $d A u $. $d B u $. $d C u $. $d R u $. $d V u $. $d W u $. $( ` B ` and ` C ` are cosets by intersection with the restricted converse epsilon class: a binary relation. (Contributed by Peter Mazsa, 31-Dec-2021.) $) br1cossincnvepres $p |- ( ( B e. V /\ C e. W ) -> ( B ,~ ( R i^i ( `' _E |` A ) ) C <-> E. u e. A ( ( B e. u /\ u R B ) /\ ( C e. u /\ u R C ) ) ) ) $= ( wcel wa cep ccnv cres cin wbr wrex wb cvv brcnvep elv anbi1i cv anbi12i ccoss br1cossinres rexbii syl6bb ) CFHDGHICDEJKZBLMUCNAUAZCUGNZUHCENZIZUH DUGNZUHDENZIZIZABOCUHHZUJIZDUHHZUMIZIZABOABCDEUGFGUDUOUTABUKUQUNUSUIUPUJU IUPPAUHCQRSTULURUMULURPAUHDQRSTUBUEUF $. $} ${ $d A u $. $d B u $. $d C u $. $d D u $. $d E u $. $d R u $. $d V u $. $d W u $. $d X u $. $d Y u $. $( ` <. B , C >. ` and ` <. D , E >. ` are cosets by tail Cartesian product with the restricted identity class: a binary relation. (Contributed by Peter Mazsa, 8-Jun-2021.) $) br1cosstxpidres $p |- ( ( ( B e. V /\ C e. W ) /\ ( D e. X /\ E e. Y ) ) -> ( <. B , C >. ,~ ( R (x) ( _I |` A ) ) <. D , E >. <-> E. u e. A ( ( u = C /\ u R B ) /\ ( u = E /\ u R D ) ) ) ) $= ( wcel wa cop cid wbr wrex wceq wb cvv cres ccoss br1cosstxpres ideq2 elv ctxp cv anbi1i anbi12i rexbii syl6bb ) CHLDILMEJLGKLMMCDNEGNFOBUAUFUBPAUG ZDOPZULCFPZMZULGOPZULEFPZMZMZABQULDRZUNMZULGRZUQMZMZABQABCDEFOGHIJKUCUSVD ABUOVAURVCUMUTUNUMUTSAULDTUDUEUHUPVBUQUPVBSAULGTUDUEUHUIUJUK $. $} ${ $d A u $. $d B u $. $d C u $. $d D u $. $d E u $. $d R u $. $d V u $. $d W u $. $d X u $. $d Y u $. $( ` <. B , C >. ` and ` <. D , E >. ` are cosets by tail Cartesian product with the restricted converse epsilon class: a binary relation. (Contributed by Peter Mazsa, 12-May-2021.) $) br1cosstxpcnvepres $p |- ( ( ( B e. V /\ C e. W ) /\ ( D e. X /\ E e. Y ) ) -> ( <. B , C >. ,~ ( R (x) ( `' _E |` A ) ) <. D , E >. <-> E. u e. A ( ( C e. u /\ u R B ) /\ ( E e. u /\ u R D ) ) ) ) $= ( wcel wa cop wbr wrex wb cvv brcnvep elv ccnv cres ctxp cv br1cosstxpres cep ccoss anbi1i anbi12i rexbii syl6bb ) CHLDILMEJLGKLMMCDNEGNFUFUAZBUBUC UGOAUDZDULOZUMCFOZMZUMGULOZUMEFOZMZMZABPDUMLZUOMZGUMLZURMZMZABPABCDEFULGH IJKUEUTVEABUPVBUSVDUNVAUOUNVAQAUMDRSTUHUQVCURUQVCQAUMGRSTUHUIUJUK $. $} $( The domain of cosets is the domain of converse. (Contributed by Peter Mazsa, 4-Jan-2019.) $) dmcoss3 $p |- dom ,~ R = dom `' R $= ( ccoss cdm ccnv ccom dfcoss3 dmeqi crn wss wceq ssid rncnv sseq1i dmcosseq mpbir ax-mp eqtri ) ABZCAADZEZCZSCZRTAFGSHZACZIZUAUBJUEUDUDIUDKUCUDUDALMOAS NPQ $. $( The domain of cosets is the range. (Contributed by Peter Mazsa, 27-Dec-2018.) $) dmcoss2 $p |- dom ,~ R = ran R $= ( ccoss cdm ccnv crn dmcoss3 df-rn eqtr4i ) ABCADCAEAFAGH $. ${ $d R x y $. $( The range of cosets is the domain of them (this should be ~ rncoss but there exists a theorem with this name already). (Contributed by Peter Mazsa, 12-Dec-2019.) $) rncossdmcoss $p |- ran ,~ R = dom ,~ R $= ( vy vx cv ccoss wbr wex cab crn cdm brcosscnvcoss el2v exbii abbii dfrn2 wb cvv df-dm 3eqtr4i ) BDZCDZAEZFZBGZCHUATUBFZBGZCHUBIUBJUDUFCUCUEBUCUEPB CTUAAQQKLMNBCUBOCBUBRS $. $} $( The domain of cosets of the converse epsilon relation restricted is the union of the restriction. (Contributed by Peter Mazsa, 18-May-2019.) (Revised by Peter Mazsa, 26-Sep-2021.) $) dm1cosscnvepres $p |- dom ,~ ( `' _E |` A ) = U. A $= ( cep ccnv cres ccoss cdm crn cuni dmcoss2 rncnvepres eqtri ) BCADZEFLGAHLI AJK $. $( The domain of coelements in ` A ` is the union of ` A ` . (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Peter Mazsa, 5-Apr-2018.) (Revised by Peter Mazsa, 26-Sep-2021.) $) dmcoels $p |- dom ~ A = U. A $= ( ccoels cdm cep ccnv cres ccoss cuni df-coels dmeqi dm1cosscnvepres eqtri ) ABZCDEAFGZCAHMNAIJAKL $. ${ $d A u $. $d R u $. $d V u $. $( Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 29-Mar-2019.) $) eldmcoss $p |- ( A e. V -> ( A e. dom ,~ R <-> E. u u R A ) ) $= ( ccoss cdm wcel ccnv cv wbr wex dmcoss3 eleq2i eldmcnv syl5bb ) BCEFZGBC HFZGBDGAIBCJAKPQBCLMABCDNO $. $} ${ $d A u $. $d R u $. $d V u $. $( Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 28-Dec-2018.) $) eldmcoss2 $p |- ( A e. V -> ( A e. dom ,~ R <-> A ,~ R A ) ) $= ( vu wcel ccoss cdm cv wbr wex eldmcoss wa wb brcoss anidms exbii syl6bbr pm4.24 bitr4d ) ACEZABFZGEDHABIZDJZAAUAIZDABCKTUDUBUBLZDJZUCTUDUFMDAABCCN OUBUEDUBRPQS $. $} ${ $d A u $. $d B u $. $d R u $. $d V u $. $( Elementhood in the domain of restricted cosets. (Contributed by Peter Mazsa, 30-Dec-2018.) $) eldm1cossres $p |- ( B e. V -> ( B e. dom ,~ ( R |` A ) <-> E. u e. A u R B ) ) $= ( wcel cres ccoss cdm cv wbr wa wex wrex eldmcoss brresALTV exbidv df-rex bitrd syl6bbr ) CEFZCDBGZHIFZAJZBFUDCDKZLZAMZUEABNUAUCUDCUBKZAMUGACUBEOUA UHUFABUDCDEPQSUEABRT $. $} ${ $d A x $. $d B x $. $d R x $. $d V x $. $( Elementhood in the domain of restricted cosets. (Contributed by Peter Mazsa, 30-Dec-2018.) $) eldm1cossres2 $p |- ( B e. V -> ( B e. dom ,~ ( R |` A ) <-> E. x e. A B e. [ x ] R ) ) $= ( wcel cres ccoss cdm cv wbr wrex cec eldm1cossres elecALTV el2v1 rexbidv wb cvv bitr4d ) CEFZCDBGHIFAJZCDKZABLCUBDMFZABLABCDENUAUDUCABUAUDUCRAUBCD SEOPQT $. $} $( Lemma for the left side of the ~ refrelcoss3 reflexivity theorem. (Contributed by Peter Mazsa, 1-Apr-2019.) $) refrelcosslem $p |- A. x e. dom ,~ R x ,~ R x $= ( cv ccoss cdm wcel wral wbr ralid wb cvv eldmcoss2 elv ralbii mpbi ) ACZBD ZEZFZARGPPQHZARGARISTARSTJAPBKLMNO $. ${ $d R x y $. $( The class of cosets by ` R ` is reflexive, cf ~ dfrefrel3 . (Contributed by Peter Mazsa, 30-Jul-2019.) $) refrelcoss3 $p |- ( A. x e. dom ,~ R A. y e. ran ,~ R ( x = y -> x ,~ R y ) /\ Rel ,~ R ) $= ( cv wceq ccoss wbr crn wral cdm refrelcosslem idinxpssinxp4 rncossdmcoss wi wrel mpbir raleqi ralbii relcoss pm3.2i ) ADZBDZEUAUBCFZGNZBUCHZIZAUCJ ZIZUCOUHUDBUGIZAUGIZUJUAUAUCGAUGIACKABUGUCLPUFUIAUGUDBUEUGCMQRPCST $. $} ${ $d R x y $. $( The class of cosets by ` R ` is reflexive, cf ~ dfrefrel2 . (Contributed by Peter Mazsa, 30-Jul-2019.) $) refrelcoss2 $p |- ( ( _I i^i ( dom ,~ R X. ran ,~ R ) ) C_ ,~ R /\ Rel ,~ R ) $= ( vx vy cid ccoss cdm crn cxp cin wss wrel wa cv wceq wi wral refrelcoss3 wbr idinxpss anbi1i mpbir ) DAEZFZUBGZHIUBJZUBKZLBMZCMZNUGUHUBROCUDPBUCPZ UFLBCAQUEUIUFBCUCUDUBSTUA $. $} $( The class of cosets by ` R ` is symmetric, cf ~ dfsymrel3 . (Contributed by Peter Mazsa, 28-Mar-2019.) (Revised by Peter Mazsa, 17-Sep-2021.) $) symrelcoss3 $p |- ( A. x A. y ( x ,~ R y -> y ,~ R x ) /\ Rel ,~ R ) $= ( cv ccoss wbr wi wal wrel wb brcosscnvcoss el2v biimpi gen2 relcoss pm3.2i cvv ) ADZBDZCEZFZSRTFZGZBHAHTIUCABUAUBUAUBJABRSCQQKLMNCOP $. ${ $d R x y $. $( The class of cosets by ` R ` is symmetric, cf ~ dfsymrel2 . (Contributed by Peter Mazsa, 27-Dec-2018.) $) symrelcoss2 $p |- ( `' ,~ R C_ ,~ R /\ Rel ,~ R ) $= ( vx vy ccoss ccnv wss wrel wa cv wbr wal symrelcoss3 cnvsym anbi1i mpbir wi ) ADZEQFZQGZHBIZCIZQJUATQJPCKBKZSHBCALRUBSBCQMNO $. $} $( Lemma for ~ cnvrefrelcoss2 . (Contributed by Peter Mazsa, 27-Jul-2021.) $) cossssid $p |- ( ,~ R C_ _I <-> ,~ R C_ ( _I i^i ( dom ,~ R X. ran ,~ R ) ) ) $= ( ccoss cid wss cdm crn cxp wceq iss2 wrel refrelcoss2 simpli eqss mpbiran2 cin bitri ) ABZCDQCQEQFGOZHZQRDZQISTRQDZUAQJAKLQRMNP $. ${ $d R u x y $. $( Lemma for the alternate definition of the function relation predicate ~ dffunALTV3 via ~ cossssid3 . (Contributed by Peter Mazsa, 10-Mar-2019.) $) cossssid2 $p |- ( ,~ R C_ _I <-> A. x A. y ( E. u ( u R x /\ u R y ) -> x = y ) ) $= ( ccoss cid wss cv wceq copab wbr wa wex wi dfid3 sseq2i df-coss ssopab2b wal sseq1i 3bitri ) DEZFGUBAHZBHZIZABJZGCHZUCDKUGUDDKLCMZABJZUFGUHUENBSAS FUFUBABOPUBUIUFABCDQTUHUEABRUA $. $} ${ $d R u x y $. $( Lemma for the alternate definition of the function relation predicate ~ dffunALTV3 . (Contributed by Peter Mazsa, 10-Mar-2019.) $) cossssid3 $p |- ( ,~ R C_ _I <-> A. u A. x A. y ( ( u R x /\ u R y ) -> x = y ) ) $= ( ccoss cid wss cv wbr wa wex wceq wi cossssid2 19.23v albii alcom bitr3i wal 3bitri ) DEFGCHZAHZDIUABHZDIJZCKUBUCLZMZBSZASUDUEMZBSZCSZASUIASCSABCD NUGUJAUGUHCSZBSUJUKUFBUDUECOPUHBCQRPUIACQT $. $} ${ $d R u x y $. $( Lemma for the alternate definition of the function relation predicate ~ dffunALTV4 . (Contributed by Peter Mazsa, 31-Aug-2021.) $) cossssid4 $p |- ( ,~ R C_ _I <-> A. u E* x u R x ) $= ( vy ccoss cid wss cv wbr wa wceq wi wal wmo cossssid3 breq2 albii bitr4i mo4 ) CEFGBHZAHZCIZTDHZCIZJUAUCKLDMAMZBMUBANZBMADBCOUFUEBUBUDADUAUCTCPSQR $. $} ${ $d R u x y $. $( Lemma for the alternate definition of the function relation predicate ~ dffunALTV5 . (Contributed by Peter Mazsa, 5-Sep-2021.) $) cossssid5 $p |- ( ,~ R C_ _I <-> A. x e. ran R A. y e. ran R ( x = y \/ ( [ x ] `' R i^i [ y ] `' R ) = (/) ) ) $= ( vu ccoss cid wss cv wbr wmo wal wceq ccnv cec cin c0 crn wral cossssid4 wo ineccnvmo2 bitr4i ) CEFGDHAHZCIAJDKUCBHZLUCCMZNUDUENOPLTBCQZRAUFRADCSA BDCUAUB $. $} ${ $d A x $. $d B x $. $d R x $. $d V x $. $d W x $. $( ` A ` and ` B ` are cosets by converse ` R ` : a binary relation. (Contributed by Peter Mazsa, 23-Jan-2019.) $) brcosscnv $p |- ( ( A e. V /\ B e. W ) -> ( A ,~ `' R B <-> E. x ( A R x /\ B R x ) ) ) $= ( wcel wa ccnv ccoss wbr cv wex brcoss cvv brcnvg el2v1 bi2anan9 exbidv wb bitrd ) BEGZCFGZHZBCDIZJKALZBUEKZUFCUEKZHZAMBUFDKZCUFDKZHZAMABCUEEFNUD UIULAUBUGUJUCUHUKUBUGUJTAUFBOEDPQUCUHUKTAUFCOFDPQRSUA $. $} ${ $d A x $. $d B x $. $d R x $. $d V x $. $d W x $. $( ` A ` and ` B ` are cosets by converse ` R ` : a binary relation. (Contributed by Peter Mazsa, 12-Mar-2019.) $) brcosscnv2 $p |- ( ( A e. V /\ B e. W ) -> ( A ,~ `' R B <-> ( [ A ] R i^i [ B ] R ) =/= (/) ) ) $= ( vx wcel wa ccnv ccoss wbr cv wex cec cin c0 wne brcosscnv ecinn0 bitr4d ) ADGBEGHABCIJKAFLZCKBUACKHFMACNBCNOPQFABCDERFABCDEST $. $} ${ $d A x y $. $d B x y $. $d R x y $. $d S x y $. $d V x y $. $d W x y $. $( ` A ` and ` B ` are cosets by converse tail Cartesian product: a binary relation. (Contributed by Peter Mazsa, 19-Apr-2020.) (Revised by Peter Mazsa, 21-Sep-2021.) $) br1cosscnvtxp $p |- ( ( A e. V /\ B e. W ) -> ( A ,~ `' ( R (x) S ) B <-> ( A ,~ `' R B /\ A ,~ `' S B ) ) ) $= ( vx vy wcel wa ccnv ccoss wbr cv wex cec cin c0 wne copab ctxp ineqan12d brcosscnv2 ectxp inopab syl6eq an4 neeq1d opabn0 eeanv bitri syl6bb bitrd opabbii brcosscnv anbi12d bitr4d ) AEIZBFIZJZABCDUAZKLMZAGNZCMZBVCCMZJZGO ZAHNZDMZBVHDMZJZHOZJZABCKLMZABDKLMZJUTVBAVAPZBVAPZQZRSZVMABVAEFUCUTVSVFVK JZGHTZRSZVMUTVRWARUTVRVDVIJZVEVJJZJZGHTZWAUTVRWCGHTZWDGHTZQWFURUSVPWGVQWH GHACDEUDGHBCDFUDUBWCWDGHUEUFWEVTGHVDVIVEVJUGUNUFUHWBVTHOGOVMVTGHUIVFVKGHU JUKULUMUTVNVGVOVLGABCEFUOHABDEFUOUPUQ $. $} ${ $d A x y $. $d B x y $. $( Cosets by converse tail Cartesian product. (Contributed by Peter Mazsa, 19-Apr-2020.) (Revised by Peter Mazsa, 21-Sep-2021.) $) 1cosscnvtxp $p |- ,~ `' ( A (x) B ) = ( ,~ `' A i^i ,~ `' B ) $= ( vx vy cv ctxp ccnv ccoss wbr copab cin wa wb br1cosscnvtxp wrel relcoss cvv wceq dfrel4v mpbi el2v opabbii inopab eqtr4i ineq12i 3eqtr4i ) CEZDEZ ABFGZHZIZCDJZUGUHAGZHZIZCDJZUGUHBGZHZIZCDJZKZUJUNURKULUOUSLZCDJVAUKVBCDUK VBMCDUGUHABQQNUAUBUOUSCDUCUDUJOUJULRUIPCDUJSTUNUPURUTUNOUNUPRUMPCDUNSTURO URUTRUQPCDURSTUEUF $. $} ${ $d R u v x $. $( Lemma for alternate definition of the disjoint relation predicate, cf. ~ dfdisjALTV3 . (Contributed by Peter Mazsa, 28-Jul-2021.) $) cosscnvssid3 $p |- ( ,~ `' R C_ _I <-> A. u A. v A. x ( ( u R x /\ v R x ) -> u = v ) ) $= ( ccnv ccoss cid wss cv wbr wa wceq wi wal cossssid3 alrot3 wb cvv brcnvg el2v anbi12i imbi1i 3albii 3bitri ) DEZFGHAIZCIZUEJZUFBIZUEJZKZUGUILZMZBN CNANUMANBNCNUGUFDJZUIUFDJZKZULMZANBNCNCBAUEOUMACBPUMUQCBAUKUPULUHUNUJUOUH UNQACUFUGRRDSTUJUOQABUFUIRRDSTUAUBUCUD $. $} ${ $d R u x $. $( Lemma for alternate definition of the disjoint relation predicate, cf. ~ dfdisjALTV4 . (Contributed by Peter Mazsa, 31-Aug-2021.) $) cosscnvssid4 $p |- ( ,~ `' R C_ _I <-> A. x E* u u R x ) $= ( ccnv ccoss cid wss cv wbr wmo wal cossssid4 cvv brcnvg el2v mobii albii wb bitri ) CDZEFGAHZBHZTIZBJZAKUBUACIZBJZAKBATLUDUFAUCUEBUCUERABUAUBMMCNO PQS $. $} ${ $d R u v x $. $( Lemma for ~ dfdisjs5 , ~ dfdisjALTV5 , and ~ eldisjs5 . (Contributed by Peter Mazsa, 5-Sep-2021.) $) cosscnvssid5 $p |- ( ( ,~ `' R C_ _I /\ Rel R ) <-> ( A. u e. dom R A. v e. dom R ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) /\ Rel R ) ) $= ( vx ccnv ccoss cid wss wrel wa cv wbr wmo wal wceq cec cin c0 wo wral cdm cosscnvssid4 anbi1i inecmo3 bitr4i ) CEFGHZCIZJBKZDKCLBMDNZUGJUHAKZOU HCPUJCPQROSACUAZTBUKTUGJUFUIUGDBCUBUCDABCUDUE $. $} ${ $d x y z $. $( Cosets by the empty set are the empty set. (Contributed by Peter Mazsa, 22-Oct-2019.) $) coss0 $p |- ,~ (/) = (/) $= ( vy vx vz c0 ccoss cv cec wcel wa wex copab dfcoss2 eleq2i anbi12i exbii ec0 nfv wn cvv mpbir 19.9 bitri opabbii cpr wss wne prnzg ss0b necon3bbii elv wb prssg el2v notbii opabf 3eqtri ) DEAFZBFZDGZHZCFZUSHZIZBJZACKUQDHZ VADHZIZACKDACBDLVDVGACVDVGBJVGVCVGBUTVEVBVFUSDUQURPZMUSDVAVHMNOVGBVGBQUAU BUCVGACVGRUQVAUDZDUEZRZVKVIDUFZVLAUQVASUGUJVJVIDVIUHUITVGVJVGVJUKACUQVADS SULUMUNTUOUP $. $} ${ $d x y z $. $( Cosets by the identity relation are the identity relation. (Contributed by Peter Mazsa, 16-Jan-2019.) $) cossid $p |- ,~ _I = _I $= ( vy vz vx cv copab cid wbr wa wex ccoss wb cvv eqvincg elv ideqg anbi12i wceq exbii bitr4i opabbii dfid3 df-coss 3eqtr4ri ) ADZBDZQZABECDZUDFGZUGU EFGZHZCIZABEFFJUFUKABUFUGUDQZUGUEQZHZCIZUKUFUOKACUDUELMNUJUNCUHULUIUMUHUL KAUGUDLONUIUMKBUGUELONPRSTABUAABCFUBUC $. $} $( Cosets by the converse identity relation are the identity relation. (Contributed by Peter Mazsa, 27-Sep-2021.) $) cosscnvid $p |- ,~ `' _I = _I $= ( cid ccnv ccoss cnvi cosseqi cossid eqtri ) ABZCACAHADEFG $. ${ $d R u x $. $d R u z $. $d u x y $. $d y z $. $( Sufficient condition for transitivity of cosets by ` R ` . (Contributed by Peter Mazsa, 26-Dec-2018.) $) trcoss $p |- ( A. y E* u u R y -> A. x A. y A. z ( ( x ,~ R y /\ y ,~ R z ) -> x ,~ R z ) ) $= ( cv wbr wmo wal wa wi nfv 19.3 wex wb cvv brcoss el2v alimi sylbir ccoss motr anbi12i imbi12i sylibr ) DFZBFZEGZDHZBIZUJAIAFZUGEUAZGZUGCFZULGZJZUK UNULGZKZCIZBIZAIUJAUJALMUJUTAUIUSBUIUICIUSUICUICLMUIURCUIUFUKEGZUHJDNZUHU FUNEGZJDNZJZVAVCJDNZKURVAUHVCDUBUPVEUQVFUMVBUOVDUMVBOABDUKUGEPPQRUOVDOBCD UGUNEPPQRUCUQVFOACDUKUNEPPQRUDUESTSST $. $} $( Lemma for expressing equivalent cosets ( ~ eqvrelcoss4 via ~ trcoss2 ). (Contributed by Peter Mazsa, 15-Oct-2021.) $) eleccossin $p |- ( ( B e. V /\ C e. W ) -> ( B e. ( [ A ] ,~ R i^i [ C ] ,~ R ) <-> ( A ,~ R B /\ B ,~ R C ) ) ) $= ( wcel wa ccoss wbr cec cin brcosscnvcoss anbi2d elin wrel wb relcoss ax-mp relelec anbi12i bitri syl6rbbr ) BEGCFGHZABDIZJZBCUEJZHUFCBUEJZHZBAUEKZCUEK ZLGZUDUGUHUFBCDEFMNULBUJGZBUKGZHUIBUJUKOUMUFUNUHUEPZUMUFQDRZBAUETSUOUNUHQUP BCUETSUAUBUC $. ${ $d R y $. $d x y $. $d y z $. $( Lemma for expressing equivalent cosets, cf. ~ eqvrelcoss4 . (Contributed by Peter Mazsa, 4-Jul-2020.) (Revised by Peter Mazsa, 16-Oct-2021.) $) trcoss2 $p |- ( A. x A. y A. z ( ( x ,~ R y /\ y ,~ R z ) -> x ,~ R z ) <-> A. x A. z ( ( [ x ] ,~ R i^i [ z ] ,~ R ) =/= (/) -> ( [ x ] `' R i^i [ z ] `' R ) =/= (/) ) ) $= ( cv ccoss wbr wa wi wal cec cin c0 wne ccnv alcom albii wb cvv el2v wcel wex 19.23v eleccossin bicomi brcoss3 imbi12i imbi1i 3bitr4i 2albii bitri n0 ) AEZBEZDFZGUNCEZUOGHZUMUPUOGZIZCJBJZAJUSBJZCJZAJUMUOKUPUOKLZMNZUMDOZK UPVEKLMNZIZCJAJUTVBAUSBCPQVAVGACUNVCUAZVFIZBJVHBUBZVFIVAVGVHVFBUCUSVIBUQV HURVFVHUQVHUQRBCUMUNUPDSSUDTUEURVFRACUMUPDSSUFTUGQVDVJVFBVCULUHUIUJUK $. $} $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Relations =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) $( Define the relations class. Proper class relations (like ` _I ` , cf. ~ reli ) are not elements of it. The element of this class and the relation predicate are the same when ` R ` is a set (cf.) ~ elrelsrel ). (Contributed by Peter Mazsa, 13-Jun-2018.) $) df-rels $a |- Rels = ~P ( _V X. _V ) $. $( The element of the relations class ( ~ df-rels ) and the relation predicate ( ~ df-rel ) are the same when ` R ` is a set. (Cf. ~ elfunsALTV2 , ~ eldisjs2 .) (Contributed by Peter Mazsa, 14-Jun-2018.) $) elrels2 $p |- ( R e. V -> ( R e. Rels <-> R C_ ( _V X. _V ) ) ) $= ( crels wcel cvv cxp cpw wss df-rels eleq2i elpwg syl5bb ) ACDAEEFZGZDABDAM HCNAIJAMBKL $. $( The element of the relations class ( ~ df-rels ) and the relation predicate are the same when ` R ` is a set. (Contributed by Peter Mazsa, 24-Nov-2018.) $) elrelsrel $p |- ( R e. V -> ( R e. Rels <-> Rel R ) ) $= ( wcel crels cvv cxp wss wrel elrels2 df-rel syl6bbr ) ABCADCAEEFGAHABIAJK $. $( The element of the relations class is a relation. (Contributed by Peter Mazsa, 20-Jul-2019.) $) elrelsrelim $p |- ( R e. Rels -> Rel R ) $= ( crels wcel wrel elrelsrel ibi ) ABCADABEF $. $( Element of the relations class. (Contributed by Peter Mazsa, 21-Jul-2021.) $) elrels5 $p |- ( R e. V -> ( R e. Rels <-> ( R |` dom R ) = R ) ) $= ( wcel crels wrel cdm cres wceq elrelsrel dfrel5 syl6bb ) ABCADCAEAAFGAHABI AJK $. $( Element of the relations class. (Contributed by Peter Mazsa, 21-Jul-2021.) $) elrels6 $p |- ( R e. V -> ( R e. Rels <-> ( R i^i ( dom R X. ran R ) ) = R ) ) $= ( wcel crels wrel cdm crn cxp cin wceq elrelsrel dfrel6 syl6bb ) ABCADCAEAA FAGHIAJABKALM $. ${ $d R x y $. $( Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.) $) twsymr2 $p |- ( R e. Rels -> ( R = `' R <-> A. x A. y ( x R y -> y R x ) ) ) $= ( ccnv wceq wss wa crels wcel cv wbr wi wal eqss cnvsym biimpi a1d adantl com12 wrel elrelsrelim dfrel2 sylib cnvss sseq1 syl5ibcom syl imp biimpri sylbir jca ex impbid syl5bb ) CCDZECUOFZUOCFZGZCHIZAJZBJZCKVAUTCKLBMAMZCU ONUSURVBURUSVBUQUSVBLUPUQVBUSUQVBABCOZPQRSUSVBURUSVBGUPUQUSVBUPUSUODZCEZV BUPLUSCTVECUACUBUCVBVEUPVBUQVEUPLVCUQVDUOFVEUPUOCUDVDCUOUEUFUJSUGUHVBUQUS UQVBVCUIRUKULUMUN $. $} ${ $d R x y $. $( Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.) $) elrelscnveq $p |- ( R e. Rels -> ( `' R C_ R <-> `' R = R ) ) $= ( vx vy ccnv wceq crels wcel wss eqcom cv wbr wal twsymr2 cnvsym syl5rbbr wi syl6bbr ) ADZAEAREZAFGZRAHZARITSBJZCJZAKUCUBAKPCLBLUABCAMBCANQO $. $} ${ $d R x y $. $( Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.) $) elrelscnveq2 $p |- ( R e. Rels -> ( `' R = R <-> A. x A. y ( x R y <-> y R x ) ) ) $= ( crels wcel ccnv wss wa cv wbr wal wceq cnvsym a1i elrelsrelim relbrcnvg wi wb wrel syl dfrel2 sylib sseq1d syl5rbbr imbi12d 2albidv bitrd anbi12d eqss 2albiim 3bitr4g ) CDEZCFZCGZCUMGZHAIZBIZCJZUQUPCJZQBKAKZUSURQZBKAKZH UMCLURUSRBKAKULUNUTUOVBUNUTRULABCMNULUOUPUQUMJZUQUPUMJZQZBKAKZVBVFUMFZUMG ULUOABUMMULVGCUMULCSZVGCLCOZCUAUBUCUDULVEVAABULVCUSVDURULVHVCUSRVIUPUQCPT ULVHVDURRVIUQUPCPTUEUFUGUHUMCUIURUSABUJUK $. $} ${ $d R x y $. $( Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.) $) elrelscnveq3 $p |- ( R e. Rels -> ( `' R C_ R <-> A. x A. y ( x R y <-> y R x ) ) ) $= ( crels wcel ccnv wss wceq cv wbr wb wal elrelscnveq elrelscnveq2 bitrd ) CDECFZCGPCHAIZBIZCJRQCJKBLALCMABCNO $. $} $( The converse of a set is an element of the class of relations. (Contributed by Peter Mazsa, 18-Aug-2019.) $) cnvelrels $p |- ( A e. V -> `' A e. Rels ) $= ( wcel ccnv crels wrel relcnv cvv wb cnvexg elrelsrel syl mpbiri ) ABCZADZE CZOFZAGNOHCPQIABJOHKLM $. $( Cosets of sets are elements of the relations class. Implies ` |- ( R e. Rels -> ,~ R e. Rels ) ` . (Contributed by Peter Mazsa, 25-Aug-2021.) $) cosselrels $p |- ( A e. V -> ,~ A e. Rels ) $= ( wcel ccoss cvv crels cossex wrel relcoss elrelsrel mpbiri syl ) ABCADZECZ MFCZABGNOMHAIMEJKL $. $( Cosets of converse sets are elements of the relations class. (Contributed by Peter Mazsa, 31-Aug-2021.) $) cosscnvelrels $p |- ( A e. V -> ,~ `' A e. Rels ) $= ( wcel ccnv crels ccoss cnvelrels cosselrels syl ) ABCADZECJFECABGJEHI $. $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Subset relations =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) ${ $d x y $. $( Define the subsets class or the class of all subset relations. Similar to definitions of epsilon relation ( ~ df-eprel ) and identity relation ( ~ df-id ) classes. The subset relation and subclass relationship ( ~ df-ss ) are the same, that is, ` ( A _S B <-> A C_ B ) ` when ` B ` is a set, cf. ~ brssr . Subset relation class and Scott Fenton's subset class ~ df-sset are the same, cf. ~ ssreqsset . We need ` _S ` to define reflexive classes so that we can define the converse reflexive classes with the help of the converse of ` _S ` , cf. ~ df-refrels , ~ df-refrel , ~ elrefrelsrel vs. ~ df-cnvrefrels , ~ df-cnvrefrel , ~ elcnvrefrelsrel . The subsets class ` _S ` has another place in set.mm as well: if we define extensional relation based on the common property in ~ extid , ~ extep and ~ extssr , then extrelssr $p |- ExtRel _S is a theorem along with extrelep $p |- ExtRel _E and extrelid $p |- ExtRel _I . df-ss is an alternative tag for the class of subsets ` _S ` since the definition of subclass relation ` A C_ B ` , ~ df-ss should be df-sc : please consider renaming it/them for consistency. (Contributed by Peter Mazsa, 25-Jul-2019.) $) df-ssr $a |- _S = { <. x , y >. | x C_ y } $. $} ${ $d x y z $. $( Alternate definition of the subset relation. (Contributed by Peter Mazsa, 9-Aug-2021.) $) dfssr2 $p |- _S = ( ( _V X. _V ) \ ran ( _E (x) ( _V \ _E ) ) ) $= ( vz vx vy cv cep wbr cvv cdif wa wex wn copab wss cxp ctxp wcel wb epelg elv notbii crn cssr brvdif bitri anbi12i exbii dfss6 bitr4i opabbii rntxp difeq2i vvdifopab eqtri df-ssr 3eqtr4ri ) ADZBDZEFZUPCDZGEHZFZIZAJZKZBCLZ UQUSMZBCLGGNZEUTOUAZHZUBVDVFBCVDUPUQPZUPUSPZKZIZAJZKVFVCVNVBVMAURVJVAVLUR VJQBUPUQGRSVAUPUSEFZKVLUPUSEUCVOVKVOVKQCUPUSGRSTUDUEUFTAUQUSUGUHUIVIVGVCB CLZHVEVHVPVGBCAEUTUJUKVCBCULUMBCUNUO $. $} $( Subset relation class and Scott Fenton's subset class ~ df-sset are the same. (Contributed by Peter Mazsa, 9-Aug-2021.) $) ssreqsset $p |- _S = SSet $= ( cssr cvv cxp cep cdif ctxp crn csset dfssr2 df-sset eqtr4i ) ABBCDBDEFGEH IJK $. ${ $d x y $. $( The subset relation is a relation. (Contributed by Peter Mazsa, 1-Aug-2019.) $) relssr $p |- Rel _S $= ( vx vy cv wss cssr df-ssr relopabi ) ACBCDABEABFG $. $} ${ $d A x y $. $d B x y $. $( The subset relation and subclass relationship ( ~ df-ss ) are the same, that is, ` ( A _S B <-> A C_ B ) ` when ` B ` is a set. (Contributed by Peter Mazsa, 31-Jul-2019.) $) brssr $p |- ( B e. V -> ( A _S B <-> A C_ B ) ) $= ( vx vy wcel cssr wbr wss cvv wrel relssr brrelex mpan adantl simpl ssexg wa jca cv simpr ancoms sseq1 sseq2 df-ssr brabg pm5.21nd ) BCFZABGHZABIZA JFZUHRZUHUIRUKUHUIUKUHGKUIUKLABGMNOUHUIPSUJUHULUJUHRUKUHABCQUJUHUASUBDTZE TZIAUNIUJDEABJCGUMAUNUCUNBAUDDEUEUFUG $. $} $( Any set is a subset of itself. (Contributed by Peter Mazsa, 1-Aug-2019.) $) brssrid $p |- ( A e. V -> A _S A ) $= ( wcel cssr wbr wss ssid brssr mpbiri ) ABCAADEAAFAGAABHI $. $( Two ways of expressing set existence. (Contributed by Peter Mazsa, 1-Aug-2019.) $) issetssr $p |- ( A e. _V <-> A _S A ) $= ( cvv wcel cssr wbr brssrid wrel relssr brrelex mpan impbii ) ABCZAADEZABFD GMLHAADIJK $. $( Restricted subset binary relation. (Contributed by Peter Mazsa, 25-Nov-2019.) $) brssrres $p |- ( C e. V -> ( B ( _S |` A ) C <-> ( B e. A /\ B C_ C ) ) ) $= ( wcel cssr cres wbr wa wss brresALTV brssr anbi2d bitrd ) CDEZBCFAGHBAEZBC FHZIPBCJZIABCFDKOQRPBCDLMN $. $( Restricted converse subset binary relation. (Contributed by Peter Mazsa, 25-Nov-2019.) $) br1cnvssrres $p |- ( B e. V -> ( B `' ( _S |` A ) C <-> ( C e. A /\ C C_ B ) ) ) $= ( cssr cres ccnv wbr wcel wa wrel wb relres relbrcnvg ax-mp brssrres syl5bb wss ) BCEAFZGHZCBSHZBDICAICBRJSKTUALEAMBCSNOACBDPQ $. $( The converse of a subset relation swaps arguments. (Contributed by Peter Mazsa, 1-Aug-2019.) $) brcnvssr $p |- ( A e. V -> ( A `' _S B <-> B C_ A ) ) $= ( cssr ccnv wbr wcel wss wrel wb relssr relbrcnvg ax-mp brssr syl5bb ) ABDE FZBADFZACGBAHDIPQJKABDLMBACNO $. $( Any set is a converse subset of itself. (Contributed by Peter Mazsa, 9-Jun-2021.) $) brcnvssrid $p |- ( A e. V -> A `' _S A ) $= ( wcel cssr ccnv wbr wss ssid brcnvssr mpbiri ) ABCAADEFAAGAHAABIJ $. ${ $d A u $. $d B u $. $d C u $. $d D u $. $d E u $. $d R u $. $d V u $. $d W u $. $d X u $. $d Y u $. $( ` <. B , C >. ` and ` <. D , E >. ` are cosets by tail Cartesian product with converse subsets class restricted: a binary relation. (Contributed by Peter Mazsa, 9-Jun-2021.) $) br1cosstxpcnvssrres $p |- ( ( ( B e. V /\ C e. W ) /\ ( D e. X /\ E e. Y ) ) -> ( <. B , C >. ,~ ( R (x) ( `' _S |` A ) ) <. D , E >. <-> E. u e. A ( ( C C_ u /\ u R B ) /\ ( E C_ u /\ u R D ) ) ) ) $= ( wcel wa cop wbr wrex wss wb cvv brcnvssr cssr ccnv cv br1cosstxpres elv cres ctxp ccoss anbi1i anbi12i rexbii syl6bb ) CHLDILMEJLGKLMMCDNEGNFUAUB ZBUFUGUHOAUCZDUMOZUNCFOZMZUNGUMOZUNEFOZMZMZABPDUNQZUPMZGUNQZUSMZMZABPABCD EFUMGHIJKUDVAVFABUQVCUTVEUOVBUPUOVBRAUNDSTUEUIURVDUSURVDRAUNGSTUEUIUJUKUL $. $} ${ $d A x $. $d B x $. $d V x $. $d W x $. $( Property of subset relation, cf. ~ extid , ~ extep and the comment of ~ df-ssr . (Contributed by Peter Mazsa, 10-Jul-2019.) $) extssr $p |- ( ( A e. V /\ B e. W ) -> ( [ A ] `' _S = [ B ] `' _S <-> A = B ) ) $= ( vx cssr ccnv cec wceq cv wbr wb wal wcel wrel relssr pm3.2i wss brssr wa releccnveq ax-mp bi2bian9 albidv ssext syl6bbr syl5bb ) AFGZHBUHHIZEJZ AFKZUJBFKZLZEMZACNZBDNZTZABIZFOZUSTUIUNLUSUSPPQEABFFUAUBUQUNUJARZUJBRZLZE MURUQUMVBEUOUKUTUPULVAUJACSUJBDSUCUDEABUEUFUG $. $} $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Reflexivity =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) $( Define the class of all reflexive sets. It is used only by ~ df-refrels . We use subset relation ` _S ` ( ~ df-ssr ) here to be able to define converse reflexivity ( ~ df-cnvrefs ), cf. the comment of ~ df-ssr . The elements of this class are not necessarily relations (vs. ~ df-refrels ). Note the similarity of the definitions ~ df-refs , ~ df-syms and ~ df-trs , cf. the comments of ~ dfrefrels2 . (Contributed by Peter Mazsa, 19-Jul-2019.) $) df-refs $a |- Refs = { x | ( _I i^i ( dom x X. ran x ) ) _S ( x i^i ( dom x X. ran x ) ) } $. $( Define the class of all reflexive relations. This is practically ~ dfrefrels2 (which reveals that ` RefRels ` can not include proper classes like ` _I ` as is elements, cf. the comments of ~ dfrefrels2 ). Another alternative definition is ~ dfrefrels3 . The element of this class and the reflexive relation predicate ( ~ df-refrel ) are the same, that is, ` ( R e. RefRels <-> RefRel R ) ` when ` A ` is a set, cf. ~ elrefrelsrel . This definition is similar to the definitions of the classes of all symmetric ( ~ df-symrels ) and transitive ( ~ df-trrels ) relations. (Contributed by Peter Mazsa, 7-Jul-2019.) $) df-refrels $a |- RefRels = ( Refs i^i Rels ) $. $( Define the reflexive relation predicate. (Read: ` R ` is a reflexive relation.) With this predicate definition we are able to prove the reflexive property of proper classes as well, e.g. ~ refrelid , cf. the comment of ~ dfrefrels2 . Alternate definitions are ~ dfrefrel2 and ~ dfrefrel3 . The element of the class of all reflexive relations ( ~ df-refrels ) and this reflexive relation predicate are the same, that is ` ( R e. RefRels <-> RefRel R ) ` when ` R ` is a set, cf. ~ elrefrelsrel . (Contributed by Peter Mazsa, 16-Jul-2021.) $) df-refrel $a |- ( RefRel R <-> ( ( _I i^i ( dom R X. ran R ) ) C_ ( R i^i ( dom R X. ran R ) ) /\ Rel R ) ) $. $( Alternate definition of the class of all reflexive relations. Proper classes (like ` _I ` , cf. ~ iprc ) are not elements of this (or any) class. (0-ary) class constants are recommended for definitions (cf. the 1. Guideline at http://us.metamath.org/ileuni/mathbox.html ), and proper classes can not be elements of class constants (if a class is an element of another class, it is not a proper class but a set, cf. ~ elex ). So if we use 0-ary constant classes as our main definitions, they are valid only for sets, not for proper classes. If we need definitions for general classes as well, we need to redefine the 0-ary classes as corresponding predicates. This is the main reason why we use predicate-type definitions like ~ df-refrel even when we have constant-type definitions like this one or ~ df-refrels . Note that while elementhood in the class of all relations cancels restriction of ` r ` in ~ dfrefrels2 , it keeps restriction of ` _I ` : this is why the very similar definitions ~ df-refs , ~ df-syms and ~ df-trs diverge when we switch from (general) sets to relations in ~ dfrefrels2 , ~ dfsymrels2 and ~ dftrrels2 . (Contributed by Peter Mazsa, 20-Jul-2019.) $) dfrefrels2 $p |- RefRels = { r e. Rels | ( _I i^i ( dom r X. ran r ) ) C_ r } $= ( cid cdm crn cxp cin cssr wbr crefrels crefs crels df-refrels df-refs wcel cv wss cvv wb inex1g elv brssr ax-mp elrels6 biimpi sseq2d syl5bb abeqinbi wceq ) BAOZCUIDEZFZUIUJFZGHZUKUIPZAIJKLAMUMUKULPZUIKNZUNULQNZUMUORUQAUIUJQS TUKULQUAUBUPULUIUKUPULUIUHZUPURRAUIQUCTUDUEUFUG $. ${ $d r x y $. $( Alternate definition of the class of all reflexive relations. Note that this is definitely not the definition we are accustomed to, e.g. ~ issref or ~ df-reflexive ` |- ( R Reflexive A <-> ( R C_ ( A X. A ) /\ A. x e. A x R x ) ) ` . It turns out that the not-surprising definition which contains ` A. x e. dom r x r x ` needs symmetry as well, cf. ~ refsymrels3 . Only when this symmetry condition holds, like in case of equivalence relations, cf. ~ dfeqvrels3 , can we write the traditional form ` A. x e. dom r x r x ` for reflexive relations. For the special case with square Cartesian product when the two forms are equivalent cf. ~ idinxpssinxp4 where ` |- ( A. x e. A A. y e. A ( x = y -> x R y ) <-> A. x e. A x R x ) ` . Cf. similar definition of the converse reflexive relations class ~ dfcnvrefrels3 . (Contributed by Peter Mazsa, 8-Jul-2019.) $) dfrefrels3 $p |- RefRels = { r e. Rels | A. x e. dom r A. y e. ran r ( x = y -> x r y ) } $= ( cid cv cdm crn cxp cin wss wceq wral crels crefrels dfrefrels2 idinxpss wbr wi rabbieq ) DCEZFZTGZHITJAEZBEZKUCUDTQRBUBLAUALCMNCOABUAUBTPS $. $} $( Alternate definition of the reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021.) $) dfrefrel2 $p |- ( RefRel R <-> ( ( _I i^i ( dom R X. ran R ) ) C_ R /\ Rel R ) ) $= ( wrefrel cid cdm crn cxp cin wrel wa df-refrel wceq dfrel6 biimpi pm5.32ri wss sseq2d bitri ) ABCADAEFZGZARGZOZAHZISAOZUBIAJUBUAUCUBTASUBTAKALMPNQ $. ${ $d R x y $. $( Alternate definition of the reflexive relation predicate. Note that this is definitely not the definition we are accustomed to in case of reflexive relations, cf. comment of ~ dfrefrels3 . (Contributed by Peter Mazsa, 8-Jul-2019.) $) dfrefrel3 $p |- ( RefRel R <-> ( A. x e. dom R A. y e. ran R ( x = y -> x R y ) /\ Rel R ) ) $= ( wrefrel cid cdm crn cxp cin wss wrel wa cv wceq wral dfrefrel2 idinxpss wbr wi anbi1i bitri ) CDECFZCGZHICJZCKZLAMZBMZNUFUGCRSBUCOAUBOZUELCPUDUHU EABUBUCCQTUA $. $} ${ $d R r $. $( Element of the class of all reflexive relations. (Contributed by Peter Mazsa, 23-Jul-2019.) $) elrefrels2 $p |- ( R e. RefRels <-> ( ( _I i^i ( dom R X. ran R ) ) C_ R /\ R e. Rels ) ) $= ( vr cid cdm crn cxp cin crels crefrels dfrefrels2 wceq dmeq rneq xpeq12d cv wss ineq2d id sseq12d rabeqel ) CBOZDZUAEZFZGZUAPCADZAEZFZGZAPBHIABJUA AKZUEUIUAAUJUDUHCUJUBUFUCUGUAALUAAMNQUJRST $. $} ${ $d R r x y $. $( Element of the class of all reflexive relations. (Contributed by Peter Mazsa, 23-Jul-2019.) $) elrefrels3 $p |- ( R e. RefRels <-> ( A. x e. dom R A. y e. ran R ( x = y -> x R y ) /\ R e. Rels ) ) $= ( vr cv wceq wbr wi crn wral cdm crels crefrels dfrefrels3 dmeq rneq breq imbi2d raleqbidv rabeqel ) AEZBEZFZUAUBDEZGZHZBUDIZJZAUDKZJUCUAUBCGZHZBCI ZJZACKZJDLMCABDNUDCFZUHUMAUIUNUDCOUOUFUKBUGULUDCPUOUEUJUCUAUBUDCQRSST $. $} $( The element of the class of all reflexive relations and the reflexive relation predicate are the same, that is ` ( R e. RefRels <-> RefRel R ) ` when ` R ` is a set. (Contributed by Peter Mazsa, 25-Jul-2021.) $) elrefrelsrel $p |- ( R e. V -> ( R e. RefRels <-> RefRel R ) ) $= ( wcel cid cdm crn cxp cin wss crels wrel crefrels wrefrel elrelsrel anbi2d wa elrefrels2 dfrefrel2 3bitr4g ) ABCZDAEAFGHAIZAJCZPUAAKZPALCAMTUBUCUAABNO AQARS $. $( Equality theorem for reflexive relation. (Contributed by Peter Mazsa, 15-Apr-2019.) (Revised by Peter Mazsa, 23-Sep-2021.) $) refreleq $p |- ( R = S -> ( RefRel R <-> RefRel S ) ) $= ( wceq cid cdm crn cxp cin wrel wa wrefrel dmeq rneq xpeq12d ineq2d sseq12d wss id releq dfrefrel2 anbi12d 3bitr4g ) ABCZDAEZAFZGZHZAQZAIZJDBEZBFZGZHZB QZBIZJAKBKUCUHUNUIUOUCUGUMABUCUFULDUCUDUJUEUKABLABMNOUCRPABSUAATBTUB $. $( Identity relation is reflexive. (Contributed by Peter Mazsa, 25-Jul-2021.) $) refrelid $p |- RefRel _I $= ( cid wrefrel cdm crn cxp cin wss wrel ssid reli df-refrel mpbir2an ) ABAAC ADEFZMGAHMIJAKL $. $( The class of cosets by ` R ` is reflexive. (Contributed by Peter Mazsa, 4-Jul-2020.) $) refrelcoss $p |- RefRel ,~ R $= ( ccoss wrefrel cid cdm crn cxp cin wss wrel wa refrelcoss2 dfrefrel2 mpbir ) ABZCDOEOFGHOIOJKALOMN $. $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Converse reflexivity =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) $( Define the class of all converse reflexive sets, cf. the comment of ~ df-ssr . It is used only by ~ df-cnvrefrels . (Contributed by Peter Mazsa, 22-Jul-2019.) $) df-cnvrefs $a |- CnvRefs = { x | ( _I i^i ( dom x X. ran x ) ) `' _S ( x i^i ( dom x X. ran x ) ) } $. $( Define the class of all converse reflexive relations. This is practically ~ dfcnvrefrels2 (which uses the traditional subclass relation ` C_ ` ) : we use converse subset relation ( ~ brcnvssr ) here to ensure the similarity to the definitions of the classes of all reflexive ( ~ df-ref ), symmetric ( ~ df-syms ) and transitive ( ~ df-trs ) sets. We use this concept to define functions ( ~ df-funsALTV , ~ df-funALTV ) and disjoints ( ~ df-disjs , ~ df-disjALTV ). The element of the class of all converse reflexive relations and the converse reflexive relation predicate are the same, that is ` ( R e. RefRels <-> RefRel R ) ` when ` R ` is a set, cf. ~ elcnvrefrelsrel . Alternate definitions are ~ dfcnvrefrels2 and ~ dfcnvrefrels3 . (Contributed by Peter Mazsa, 7-Jul-2019.) $) df-cnvrefrels $a |- CnvRefRels = ( CnvRefs i^i Rels ) $. $( Define the converse reflexive relation predicate. (Read: ` R ` is a converse reflexive relation.) Alternate definitions are ~ dfcnvrefrel2 and ~ dfcnvrefrel3 . (Contributed by Peter Mazsa, 16-Jul-2021.) $) df-cnvrefrel $a |- ( CnvRefRel R <-> ( ( R i^i ( dom R X. ran R ) ) C_ ( _I i^i ( dom R X. ran R ) ) /\ Rel R ) ) $. $( Alternate definition of the class of all converse reflexive relations. Cf. the comment of ~ dfrefrels2 . (Contributed by Peter Mazsa, 21-Jul-2021.) $) dfcnvrefrels2 $p |- CnvRefRels = { r e. Rels | r C_ ( _I i^i ( dom r X. ran r ) ) } $= ( cid cv cdm crn cxp cin cssr ccnv ccnvrefrels ccnvrefs crels df-cnvrefrels wbr wss wcel cvv wb elv ax-mp df-cnvrefs dmexg xpex inex2ALTV brcnvssr wceq rnexg elrels6 biimpi sseq1d syl5bb abeqinbi ) BACZDZUMEZFZGZUMUPGZHINZUMUQO ZAJKLMAUAUSURUQOZUMLPZUTUQQPZUSVARUPQPVCUNUOUNQPAUMQUBSUOQPAUMQUGSUCUPBQUDT UQURQUETVBURUMUQVBURUMUFZVBVDRAUMQUHSUIUJUKUL $. ${ $d r x y $. $( Alternate definition of the class of all converse reflexive relations. Cf. the comment of ~ dfrefrels3 . (Contributed by Peter Mazsa, 22-Jul-2019.) $) dfcnvrefrels3 $p |- CnvRefRels = { r e. Rels | A. x e. dom r A. y e. ran r ( x r y -> x = y ) } $= ( cid cv cdm crn cxp cin cssr ccnv wbr wceq wral ccnvrefrels cvv wcel elv crels ax-mp wi ccnvrefs df-cnvrefrels df-cnvrefs abeqin wss wb dmexg xpex rnexg inex2ALTV brcnvssr inxpssidinxp bitri rabbieq ) DCEZFZUPGZHZIZUPUSI ZJKLZAEZBEZUPLVCVDMUABURNAUQNZCSOVBCOUBSUCCUDUEVBVAUTUFZVEUTPQZVBVFUGUSPQ VGUQURUQPQCUPPUHRURPQCUPPUJRUIUSDPUKTUTVAPULTABUQURUPUMUNUO $. $} $( Alternate definition of the converse reflexive relation predicate. (Contributed by Peter Mazsa, 24-Jul-2019.) $) dfcnvrefrel2 $p |- ( CnvRefRel R <-> ( R C_ ( _I i^i ( dom R X. ran R ) ) /\ Rel R ) ) $= ( wcnvrefrel cdm crn cxp cin cid wss wrel df-cnvrefrel dfrel6 biimpi sseq1d wa wceq pm5.32ri bitri ) ABAACADEZFZGRFZHZAIZNATHZUBNAJUBUAUCUBSATUBSAOAKLM PQ $. ${ $d R x y $. $( Alternate definition of the converse reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021.) $) dfcnvrefrel3 $p |- ( CnvRefRel R <-> ( A. x e. dom R A. y e. ran R ( x R y -> x = y ) /\ Rel R ) ) $= ( wcnvrefrel cdm crn cxp cin cid wss wrel wa cv wceq wi wral df-cnvrefrel wbr inxpssidinxp anbi1i bitri ) CDCCEZCFZGZHIUDHJZCKZLAMZBMZCRUGUHNOBUCPA UBPZUFLCQUEUIUFABUBUCCSTUA $. $} ${ $d R r $. $( Element of the class of all converse reflexive relations. (Contributed by Peter Mazsa, 25-Jul-2019.) $) elcnvrefrels2 $p |- ( R e. CnvRefRels <-> ( R C_ ( _I i^i ( dom R X. ran R ) ) /\ R e. Rels ) ) $= ( vr cv cid cdm crn cxp cin wss crels ccnvrefrels dfcnvrefrels2 wceq dmeq id rneq xpeq12d ineq2d sseq12d rabeqel ) BCZDUAEZUAFZGZHZIADAEZAFZGZHZIBJ KABLUAAMZUAAUEUIUJOUJUDUHDUJUBUFUCUGUAANUAAPQRST $. $} ${ $d R r x y $. $( Element of the class of all converse reflexive relations. (Contributed by Peter Mazsa, 30-Aug-2021.) $) elcnvrefrels3 $p |- ( R e. CnvRefRels <-> ( A. x e. dom R A. y e. ran R ( x R y -> x = y ) /\ R e. Rels ) ) $= ( vr cv wbr wceq wi crn wral cdm ccnvrefrels dfcnvrefrels3 dmeq rneq breq crels imbi1d raleqbidv rabeqel ) AEZBEZDEZFZUAUBGZHZBUCIZJZAUCKZJUAUBCFZU EHZBCIZJZACKZJDQLCABDMUCCGZUHUMAUIUNUCCNUOUFUKBUGULUCCOUOUDUJUEUAUBUCCPRS ST $. $} $( The element of the class of all converse reflexive relations and the converse reflexive relation predicate are the same, that is ` ( R e. RefRels <-> RefRel R ) ` when ` R ` is a set. (Contributed by Peter Mazsa, 25-Jul-2021.) $) elcnvrefrelsrel $p |- ( R e. V -> ( R e. CnvRefRels <-> CnvRefRel R ) ) $= ( wcel cid cdm crn cxp cin crels wa ccnvrefrels wcnvrefrel elrelsrel anbi2d wss wrel elcnvrefrels2 dfcnvrefrel2 3bitr4g ) ABCZADAEAFGHOZAICZJUAAPZJAKCA LTUBUCUAABMNAQARS $. $( Necessary and sufficient condition for a coset relation to be a converse reflexive relation. (Contributed by Peter Mazsa, 27-Jul-2021.) $) cnvrefrelcoss2 $p |- ( CnvRefRel ,~ R <-> ,~ R C_ _I ) $= ( wcnvrefrel cid cdm crn cxp cin wss relcoss dfcnvrefrel2 mpbiran2 cossssid ccoss wrel bitr4i ) AMZBZPCPDPEFGHZPCHQRPNAIPJKALO $. $( Necessary and sufficient condition for a coset relation to be an element of the converse reflexive relation class. (Contributed by Peter Mazsa, 25-Aug-2021.) $) cosselcnvrefrels2 $p |- ( ,~ R e. CnvRefRels <-> ( ,~ R C_ _I /\ ,~ R e. Rels ) ) $= ( ccoss ccnvrefrels cid cdm crn cxp cin wss crels wa elcnvrefrels2 cossssid wcel anbi1i bitr4i ) ABZCNQDQEQFGHIZQJNZKQDIZSKQLTRSAMOP $. ${ $d R u x y $. $( Necessary and sufficient condition for a coset relation to be an element of the converse reflexive relation class. (Contributed by Peter Mazsa, 30-Aug-2021.) $) cosselcnvrefrels3 $p |- ( ,~ R e. CnvRefRels <-> ( A. u A. x A. y ( ( u R x /\ u R y ) -> x = y ) /\ ,~ R e. Rels ) ) $= ( ccoss ccnvrefrels wcel cid wss crels wa cv wbr wi wal cosselcnvrefrels2 wceq cossssid3 anbi1i bitri ) DEZFGUAHIZUAJGZKCLZALZDMUDBLZDMKUEUFQNBOAOC OZUCKDPUBUGUCABCDRST $. $} ${ $d R u x $. $( Necessary and sufficient condition for a coset relation to be an element of the converse reflexive relation class. (Contributed by Peter Mazsa, 31-Aug-2021.) $) cosselcnvrefrels4 $p |- ( ,~ R e. CnvRefRels <-> ( A. u E* x u R x /\ ,~ R e. Rels ) ) $= ( ccoss ccnvrefrels wcel cid wss crels wa wbr cosselcnvrefrels2 cossssid4 cv wmo wal anbi1i bitri ) CDZEFSGHZSIFZJBNANCKAOBPZUAJCLTUBUAABCMQR $. $} ${ $d R x y $. $( Necessary and sufficient condition for a coset relation to be an element of the converse reflexive relation class. (Contributed by Peter Mazsa, 5-Sep-2021.) $) cosselcnvrefrels5 $p |- ( ,~ R e. CnvRefRels <-> ( A. x e. ran R A. y e. ran R ( x = y \/ ( [ x ] `' R i^i [ y ] `' R ) = (/) ) /\ ,~ R e. Rels ) ) $= ( ccoss ccnvrefrels wcel cid wss crels wa cv wceq ccnv cec cin c0 wo wral crn cosselcnvrefrels2 cossssid5 anbi1i bitri ) CDZEFUDGHZUDIFZJAKZBKZLUGC MZNUHUINOPLQBCSZRAUJRZUFJCTUEUKUFABCUAUBUC $. $} $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Symmetry =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) $( Define the class of all symmetric sets. It is used only by ~ df-symrels . Note the similarity of the definitions ~ df-refs , ~ df-syms and ~ df-trs , cf. the comment of ~ dfrefrels2 . (Contributed by Peter Mazsa, 19-Jul-2019.) $) df-syms $a |- Syms = { x | `' ( x i^i ( dom x X. ran x ) ) _S ( x i^i ( dom x X. ran x ) ) } $. $( Define the class of all symmetric relations. The element of the class of all symmetric relations and the symmetric relation predicate are the same when ` R ` is a set, cf. ~ elsymrelsrel . Alternate definitions are ~ dfsymrels2 , ~ dfsymrels3 , ~ dfsymrels4 and ~ dfsymrels5 . This definition is similar to the definitions of the classes of all reflexive ( ~ df-refrels ) and transitive ( ~ df-trrels ) relations. (Contributed by Peter Mazsa, 7-Jul-2019.) $) df-symrels $a |- SymRels = ( Syms i^i Rels ) $. $( Define the symmetric relation predicate. (Read: ` R ` is a symmetric relation.) The element of the class of all symmetric relations and the symmetric relation predicate are the same when ` R ` is a set, cf. ~ elsymrelsrel . Alternate definitions are ~ dfsymrel2 and ~ dfsymrel3 . (Contributed by Peter Mazsa, 16-Jul-2021.) $) df-symrel $a |- ( SymRel R <-> ( `' ( R i^i ( dom R X. ran R ) ) C_ ( R i^i ( dom R X. ran R ) ) /\ Rel R ) ) $. $( Alternate definition of the class of all symmetric relations. Cf. the comment of ~ dfrefrels2 . (Contributed by Peter Mazsa, 20-Jul-2019.) $) dfsymrels2 $p |- SymRels = { r e. Rels | `' r C_ r } $= ( cdm crn cxp cin ccnv cssr wbr wss csymrels csyms crels df-symrels df-syms cv wcel cvv wb inex1g elv brssr wceq elrels6 biimpi cnveqd sseq12d abeqinbi ax-mp syl5bb ) AOZUJBUJCDZEZFZULGHZUJFZUJIZAJKLMANUNUMULIZUJLPZUPULQPZUNUQR USAUJUKQSTUMULQUAUHURUMUOULUJURULUJURULUJUBZURUTRAUJQUCTUDZUEVAUFUIUG $. ${ $d r x y $. $( Alternate definition of the class of all symmetric relations. (Contributed by Peter Mazsa, 22-Jul-2021.) $) dfsymrels3 $p |- SymRels = { r e. Rels | A. x A. y ( x r y -> y r x ) } $= ( cv ccnv wss wbr wi wal crels csymrels dfsymrels2 cnvsym rabbieq ) CDZEO FADZBDZOGQPOGHBIAICJKCLABOMN $. $} $( Alternate definition of the class of all symmetric relations. (Contributed by Peter Mazsa, 20-Jul-2019.) $) dfsymrels4 $p |- SymRels = { r e. Rels | `' r = r } $= ( cv ccnv wss wceq crels csymrels dfsymrels2 elrelscnveq rabimbieq ) ABZCZK DLKEAFGAHKIJ $. ${ $d r x y $. $( Alternate definition of the class of all symmetric relations. (Contributed by Peter Mazsa, 22-Jul-2021.) $) dfsymrels5 $p |- SymRels = { r e. Rels | A. x A. y ( x r y <-> y r x ) } $= ( cv ccnv wceq wbr wal crels csymrels dfsymrels4 elrelscnveq2 rabimbieq wb ) CDZEOFADZBDZOGQPOGNBHAHCIJCKABOLM $. $} $( Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 19-Apr-2019.) (Revised by Peter Mazsa, 17-Aug-2021.) $) dfsymrel2 $p |- ( SymRel R <-> ( `' R C_ R /\ Rel R ) ) $= ( wsymrel cdm crn cxp cin ccnv wss wrel df-symrel wceq dfrel6 biimpi cnveqd wa sseq12d pm5.32ri bitri ) ABAACADEFZGZSHZAIZOAGZAHZUBOAJUBUAUDUBTUCSAUBSA UBSAKALMZNUEPQR $. ${ $d R x y $. $( Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 21-Apr-2019.) (Revised by Peter Mazsa, 17-Aug-2021.) $) dfsymrel3 $p |- ( SymRel R <-> ( A. x A. y ( x R y -> y R x ) /\ Rel R ) ) $= ( wsymrel ccnv wss wrel wa cv wbr wi wal dfsymrel2 cnvsym anbi1i bitri ) CDCECFZCGZHAIZBIZCJTSCJKBLALZRHCMQUARABCNOP $. $} $( Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.) $) dfsymrel4 $p |- ( SymRel R <-> ( `' R = R /\ Rel R ) ) $= ( wsymrel ccnv wss wrel wa wceq dfsymrel2 relcnveq pm5.32ri bitri ) ABACZAD ZAEZFLAGZNFAHNMOAIJK $. ${ $d R x y $. $( Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.) $) dfsymrel5 $p |- ( SymRel R <-> ( A. x A. y ( x R y <-> y R x ) /\ Rel R ) ) $= ( wsymrel ccnv wss wrel wa cv wbr wal dfsymrel2 relcnveq3 pm5.32ri bitri wb ) CDCECFZCGZHAIZBIZCJTSCJPBKAKZRHCLRQUAABCMNO $. $} ${ $d R r $. $( Element of the class of all symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.) $) elsymrels2 $p |- ( R e. SymRels <-> ( `' R C_ R /\ R e. Rels ) ) $= ( vr cv ccnv wss crels csymrels dfsymrels2 wceq cnveq id sseq12d rabeqel ) BCZDZNEADZAEBFGABHNAIZOPNANAJQKLM $. $} ${ $d R r x y $. $( Element of the class of all symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.) $) elsymrels3 $p |- ( R e. SymRels <-> ( A. x A. y ( x R y -> y R x ) /\ R e. Rels ) ) $= ( vr cv wbr wi wal crels csymrels dfsymrels3 wceq imbi12d 2albidv rabeqel breq ) AEZBEZDEZFZRQSFZGZBHAHQRCFZRQCFZGZBHAHDIJCABDKSCLZUBUEABUFTUCUAUDQ RSCPRQSCPMNO $. $} ${ $d R r $. $( Element of the class of all symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.) $) elsymrels4 $p |- ( R e. SymRels <-> ( `' R = R /\ R e. Rels ) ) $= ( vr cv ccnv wceq crels csymrels dfsymrels4 cnveq id eqeq12d rabeqel ) BC ZDZMEADZAEBFGABHMAEZNOMAMAIPJKL $. $} ${ $d R r x y $. $( Element of the class of all symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.) $) elsymrels5 $p |- ( R e. SymRels <-> ( A. x A. y ( x R y <-> y R x ) /\ R e. Rels ) ) $= ( vr cv wbr wb wal crels csymrels dfsymrels5 wceq bibi12d 2albidv rabeqel breq ) AEZBEZDEZFZRQSFZGZBHAHQRCFZRQCFZGZBHAHDIJCABDKSCLZUBUEABUFTUCUAUDQ RSCPRQSCPMNO $. $} $( The element of the class of all symmetric relations and the symmetric relation predicate are the same when ` R ` is a set. (Contributed by Peter Mazsa, 17-Aug-2021.) $) elsymrelsrel $p |- ( R e. V -> ( R e. SymRels <-> SymRel R ) ) $= ( wcel ccnv wss crels wa wrel wsymrel elrelsrel anbi2d elsymrels2 dfsymrel2 csymrels 3bitr4g ) ABCZADAEZAFCZGQAHZGANCAIPRSQABJKALAMO $. $( Equality theorem for symmetric relation. (Contributed by Peter Mazsa, 15-Apr-2019.) (Revised by Peter Mazsa, 23-Sep-2021.) $) symreleq $p |- ( R = S -> ( SymRel R <-> SymRel S ) ) $= ( wceq ccnv wss wa wsymrel cnveq id sseq12d releq anbi12d dfsymrel2 3bitr4g wrel ) ABCZADZAEZAOZFBDZBEZBOZFAGBGPRUASUBPQTABABHPIJABKLAMBMN $. $( Symmetric relation implies that the domain and the range are equal. (Contributed by Peter Mazsa, 29-Dec-2021.) $) symrelim $p |- ( SymRel R -> dom R = ran R ) $= ( wsymrel cdm ccnv crn rncnv wceq wrel dfsymrel4 simplbi rneqd syl5eqr ) AB ZACADZEAEAFMNAMNAGAHAIJKL $. $( The class of cosets by ` R ` is symmetric. (Contributed by Peter Mazsa, 20-Dec-2021.) $) symrelcoss $p |- SymRel ,~ R $= ( ccoss wsymrel ccnv wss wrel wa symrelcoss2 dfsymrel2 mpbir ) ABZCKDKEKFGA HKIJ $. $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Reflexivity and symmetry =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) $( Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, cf. ~ symrefref3 . (Contributed by Peter Mazsa, 19-Jul-2018.) $) symrefref2 $p |- ( `' R C_ R -> ( ( _I i^i ( dom R X. ran R ) ) C_ R <-> ( _I |` dom R ) C_ R ) ) $= ( ccnv wss cid cdm crn cxp cres wceq rnss rncnv sseq1i biimpi idreseqidinxp cin 3syl sseq1d ) ABZACZDAEZAFZGOZDTHZASRFZUACZTUACZUBUCIRAJUEUFUDTUAAKLMTU ANPQ $. ${ $d R x y $. $( Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, cf. ~ symrefref2 . (Contributed by Peter Mazsa, 23-Aug-2021.) $) symrefref3 $p |- ( A. x A. y ( x R y -> y R x ) -> ( A. x e. dom R A. y e. ran R ( x = y -> x R y ) <-> A. x e. dom R x R x ) ) $= ( ccnv wss cid cdm crn cxp cin cres wb wi cv wbr wal wceq wral symrefref2 cnvsym idinxpss issref bibi12i imbi12i mpbi ) CDCEZFCGZCHZIJCEZFUGKCEZLZM ANZBNZCOZUMULCOMBPAPZULUMQUNMBUHRAUGRZULULCOAUGRZLZMCSUFUOUKURABCTUIUPUJU QABUGUHCUAAUGCUBUCUDUE $. $} $( Elements of the class of reflexive relations which are elements of the class of symmetric relations as well (like the elements of the class of equivalence relations ~ dfeqvrels2 ) can use the restricted version for their reflexive part (see below), not just the ` ( _I i^i ( dom r X. ran r ) ) C_ r ` version of ~ dfrefrels2 , cf. the comment of ~ dfrefrels2 . (Contributed by Peter Mazsa, 20-Jul-2019.) $) refsymrels2 $p |- ( RefRels i^i SymRels ) = { r e. Rels | ( ( _I |` dom r ) C_ r /\ `' r C_ r ) } $= ( crefrels csymrels cin cid cdm crn cxp wss crels crab ccnv cres dfrefrels2 cv wa dfsymrels2 ineq12i inrab symrefref2 pm5.32ri rabbiia2 3eqtri ) BCDEAO ZFZUDGHDUDIZAJKZUDLUDIZAJKZDUFUHPZAJKEUEMUDIZUHPZAJKBUGCUIANAQRUFUHAJSUJULA JUHUFUKUDTUAUBUC $. ${ $d r x y $. $( Elements of the class of reflexive relations which are elements of the class of symmetric relations as well (like the elements of the class of equivalence relations ~ dfeqvrels3 ) can use the ` A. x e. dom r x r x ` version for their reflexive part, not just the ` A. x e. dom r A. y e. ran r ( x = y -> x r y ) ` version of ~ dfrefrels3 , cf. the comment of ~ dfrefrels3 . (Contributed by Peter Mazsa, 22-Jul-2019.) $) refsymrels3 $p |- ( RefRels i^i SymRels ) = { r e. Rels | ( A. x e. dom r x r x /\ A. x A. y ( x r y -> y r x ) ) } $= ( cid cv cdm cres wss ccnv wa wbr wral wi wal crels crefrels csymrels cin refsymrels2 issref cnvsym anbi12i rabbieq ) DCEZFZGUDHZUDIUDHZJAEZUHUDKAU ELZUHBEZUDKUJUHUDKMBNANZJCOPQRCSUFUIUGUKAUEUDTABUDUAUBUC $. $} $( A relation which is reflexive and symmetric (like an equivalence relation) can use the restricted version for their reflexive part (see below), not just the ` ( _I i^i ( dom R X. ran R ) ) C_ R ` version of ~ dfrefrel2 , cf. the comment of ~ dfrefrels2 . (Contributed by Peter Mazsa, 23-Aug-2021.) $) refsymrel2 $p |- ( ( RefRel R /\ SymRel R ) <-> ( ( ( _I |` dom R ) C_ R /\ `' R C_ R ) /\ Rel R ) ) $= ( wrefrel wsymrel wa cid cdm crn cxp cin ccnv wrel cres dfrefrel2 dfsymrel2 wss w3a anbi12i anandi3r 3anan32 3bitr2i symrefref2 pm5.32ri anbi1i bitri ) ABZACZDZEAFZAGHIAOZAJAOZDZAKZDZEUHLAOZUJDZULDUGUIULDZUJULDZDUIULUJPUMUEUPUF UQAMANQUIULUJRUIULUJSTUKUOULUJUIUNAUAUBUCUD $. ${ $d R x y $. $( A relation which is reflexive and symmetric (like an equivalence relation) can use the ` A. x e. dom R x R x ` version for its reflexive part, not just the ` A. x e. dom R A. y e. ran R ( x = y -> x R y ) ` version of ~ dfrefrel3 , cf. the comment of ~ dfrefrels2 . (Contributed by Peter Mazsa, 23-Aug-2021.) $) refsymrel3 $p |- ( ( RefRel R /\ SymRel R ) <-> ( ( A. x e. dom R x R x /\ A. x A. y ( x R y -> y R x ) ) /\ Rel R ) ) $= ( wrefrel wsymrel wa cv wceq wbr wi crn wral cdm wrel dfrefrel3 dfsymrel3 wal w3a anbi12i anandi3r 3anan32 3bitr2i symrefref3 pm5.32ri anbi1i bitri ) CDZCEZFZAGZBGZHUJUKCIZJBCKLACMZLZULUKUJCIJBQAQZFZCNZFZUJUJCIAUMLZUOFZUQ FUIUNUQFZUOUQFZFUNUQUORURUGVAUHVBABCOABCPSUNUQUOTUNUQUOUAUBUPUTUQUOUNUSAB CUCUDUEUF $. $} ${ $d R r $. $( Elements of the class of reflexive relations which are elements of the class of symmetric relations as well (like the elements of the class of equivalence relations ~ dfeqvrels2 ) can use the restricted version for their reflexive part (see below), not just the ` ( _I i^i ( dom R X. ran R ) ) C_ R ` version of ~ dfrefrels2 , cf. the comment of ~ dfrefrels2 . (Contributed by Peter Mazsa, 22-Jul-2019.) $) elrefsymrels2 $p |- ( R e. ( RefRels i^i SymRels ) <-> ( ( ( _I |` dom R ) C_ R /\ `' R C_ R ) /\ R e. Rels ) ) $= ( vr crefrels csymrels cin wcel cid cdm cres ccnv wa crels cv refsymrels2 wss crab eleq2i wceq dmeq sseq12d reseq2d cnveq anbi12d elrab bitri bian id ) ACDEZFZGAHZIZAOZAJZAOZKZALFZUIAGBMZHZIZUQOZUQJZUQOZKZBLPZFUPUOKUHVDA BNQVCUOBALUQARZUTULVBUNVEUSUKUQAVEURUJGUQASUAVEUGZTVEVAUMUQAUQAUBVFTUCUDU EUF $. $} ${ $d R x y $. $( Elements of the class of reflexive relations which are elements of the class of symmetric relations as well (like the elements of the class of equivalence relations ~ dfeqvrels3 ) can use the ` A. x e. dom R x R x ` version for their reflexive part, not just the ` A. x e. dom R A. y e. ran R ( x = y -> x R y ) ` version of ~ dfrefrels3 , cf. the comment of ~ dfrefrels3 . (Contributed by Peter Mazsa, 22-Jul-2019.) $) elrefsymrels3 $p |- ( R e. ( RefRels i^i SymRels ) <-> ( ( A. x e. dom R x R x /\ A. x A. y ( x R y -> y R x ) ) /\ R e. Rels ) ) $= ( crefrels csymrels cin wcel cid cdm cres wss ccnv wa crels cv wbr wi wal wral elrefsymrels2 issref cnvsym anbi12i anbi1i bitri ) CDEFGHCIZJCKZCLCK ZMZCNGZMAOZUKCPAUFSZUKBOZCPUMUKCPQBRARZMZUJMCTUIUOUJUGULUHUNAUFCUAABCUBUC UDUE $. $} $( The element of the class of all reflexive and symmetric relations and the conjunction of the reflexive and symmetric relation predicates are the same when ` R ` is a set. (Contributed by Peter Mazsa, 23-Aug-2021.) $) elrefsymrelsrel $p |- ( R e. V -> ( R e. ( RefRels i^i SymRels ) <-> ( RefRel R /\ SymRel R ) ) ) $= ( crefrels csymrels cin wcel wrefrel wsymrel elin elrefrelsrel elsymrelsrel wa anbi12d syl5bb ) ACDEFACFZADFZLABFZAGZAHZLACDIQORPSABJABKMN $. $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Transitivity =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) $( Define the class of all transitive sets (vs. the transitive class defined in ~ df-tr ). It is used only by ~ df-trrels . Note the similarity of the definitions of ~ df-refs , ~ df-syms and ~ df-trs ), cf. the comments of ~ dfrefrels2 . (Contributed by Peter Mazsa, 17-Jul-2021.) $) df-trs $a |- Trs = { x | ( ( x i^i ( dom x X. ran x ) ) o. ( x i^i ( dom x X. ran x ) ) ) _S ( x i^i ( dom x X. ran x ) ) } $. $( Define the class of all transitive relations. The element of the class of all transitive relations and the transitive relation predicate are the same when ` R ` is a set, cf. ~ eltrrelsrel . Alternate definitions are ~ dftrrels2 and ~ dftrrels3 . This definition is similar to the definitions of the classes of all reflexive ( ~ df-refrels ) and symmetric ( ~ df-symrels ) relations. (Contributed by Peter Mazsa, 7-Jul-2019.) $) df-trrels $a |- TrRels = ( Trs i^i Rels ) $. $( Define the transitive relation predicate. (Read: ` R ` is a transitive relation.) The element of the class of all transitive relations and the transitive relation predicate are the same when ` R ` is a set, cf. ~ eltrrelsrel . Alternate definitions are ~ dftrrel2 and ~ dftrrel3 . (Contributed by Peter Mazsa, 17-Jul-2021.) $) df-trrel $a |- ( TrRel R <-> ( ( ( R i^i ( dom R X. ran R ) ) o. ( R i^i ( dom R X. ran R ) ) ) C_ ( R i^i ( dom R X. ran R ) ) /\ Rel R ) ) $. $( Alternate definition of the class of all transitive relations. Cf. the comment of ~ dfrefrels2 . I'd prefer to define the class of all transitive relations by using the definition of composition by [Suppes] p. 63. df-coSUP ` ( A o. B ) = { <. x , y >. | E. u ( x A u /\ u B y ) } ` as opposed to the present definition of composition ~ df-co ` ( A o. B ) = { <. x , y >. | E. u ( x B u /\ u A y ) } ` because the Suppes definition keeps the order of ` A ` , ` B ` , ` C ` , ` R ` , ` S ` , ` T ` by default in trsinxpSUP ( ( ( R i^i ( A X. B ) ) o. ( S i^i ( B X. C ) ) ) C_ ( T i^i ( A X. C ) ) <-> A. x e. A A. y e. B A. z e. C ( ( x R y /\ y S z ) -> x T z ) ) while the present definition of composition disarranges them: trsinxp ( ( ( S i^i ( B X. C ) ) o. ( R i^i ( A X. B ) ) ) C_ ( T i^i ( A X. C ) ) <-> A. x e. A A. y e. B A. z e. C ( ( x R y /\ y S z ) -> x T z ) ) This is not mission critical to me, the implication of the Suppes definition is just more aesthetic, at least in the above case. If we swap to the Suppes definition of class composition, I would define the present class of all transitive sets as df-trsSUP and I would consider to switch the definition of the class of cosets by ` R ` from the present ~ df-coss to a df-cossSUP. But perhaps there is a mathematical reason to keep the present definition of composition. (Contributed by Peter Mazsa, 21-Jul-2021.) $) dftrrels2 $p |- TrRels = { r e. Rels | ( r o. r ) C_ r } $= ( cv cdm crn cxp cin ccom cssr wbr ctrrels ctrs crels df-trrels df-trs wcel wss cvv wb inex1g elv brssr elrels6 biimpi coeq12d sseq12d syl5bb abeqinbi ax-mp wceq ) ABZUJCUJDEZFZULGZULHIZUJUJGZUJPZAJKLMANUNUMULPZUJLOZUPULQOZUNU QRUSAUJUKQSTUMULQUAUHURUMUOULUJURULUJULUJURULUJUIZURUTRAUJQUBTUCZVAUDVAUEUF UG $. ${ $d r x y z $. $( Alternate definition of the class of all transitive relations. (Contributed by Peter Mazsa, 22-Jul-2021.) $) dftrrels3 $p |- TrRels = { r e. Rels | A. x A. y A. z ( ( x r y /\ y r z ) -> x r z ) } $= ( cv ccom wss wbr wa wi wal crels ctrrels dftrrels2 cotrg rabbieq ) DEZQF QGAEZBEZQHSCEZQHIRTQHJCKBKAKDLMDNABCQQQOP $. $} $( Alternate definition of the transitive relation predicate. (Contributed by Peter Mazsa, 22-Aug-2021.) $) dftrrel2 $p |- ( TrRel R <-> ( ( R o. R ) C_ R /\ Rel R ) ) $= ( wtrrel cdm crn cxp cin ccom wss wrel df-trrel wceq dfrel6 coeq12d sseq12d wa biimpi pm5.32ri bitri ) ABAACADEFZSGZSHZAIZOAAGZAHZUBOAJUBUAUDUBTUCSAUBS ASAUBSAKALPZUEMUENQR $. ${ $d R x y z $. $( Alternate definition of the transitive relation predicate. (Contributed by Peter Mazsa, 22-Aug-2021.) $) dftrrel3 $p |- ( TrRel R <-> ( A. x A. y A. z ( ( x R y /\ y R z ) -> x R z ) /\ Rel R ) ) $= ( wtrrel ccom wss wrel wa cv wbr wi wal dftrrel2 cotrg anbi1i bitri ) DED DFDGZDHZIAJZBJZDKUACJZDKITUBDKLCMBMAMZSIDNRUCSABCDDDOPQ $. $} ${ $d R r $. $( Element of the class of all transitive relations. (Contributed by Peter Mazsa, 22-Aug-2021.) $) eltrrels2 $p |- ( R e. TrRels <-> ( ( R o. R ) C_ R /\ R e. Rels ) ) $= ( vr cv ccom wss crels ctrrels dftrrels2 wceq id coeq12d sseq12d rabeqel ) BCZNDZNEAADZAEBFGABHNAIZOPNAQNANAQJZRKRLM $. $} ${ $d R r x y z $. $( Element of the class of all transitive relations. (Contributed by Peter Mazsa, 22-Aug-2021.) $) eltrrels3 $p |- ( R e. TrRels <-> ( A. x A. y A. z ( ( x R y /\ y R z ) -> x R z ) /\ R e. Rels ) ) $= ( vr cv wbr wa wi wal ctrrels dftrrels3 wceq breq anbi12d imbi12d 2albidv crels albidv rabeqel ) AFZBFZEFZGZUBCFZUCGZHZUAUEUCGZIZCJBJZAJUAUBDGZUBUE DGZHZUAUEDGZIZCJBJZAJERKDABCELUCDMZUJUPAUQUIUOBCUQUGUMUHUNUQUDUKUFULUAUBU CDNUBUEUCDNOUAUEUCDNPQST $. $} $( The element of the class of all transitive relations and the transitive relation predicate are the same when ` R ` is a set. (Contributed by Peter Mazsa, 22-Aug-2021.) $) eltrrelsrel $p |- ( R e. V -> ( R e. TrRels <-> TrRel R ) ) $= ( wcel ccom wss crels wa ctrrels wtrrel elrelsrel anbi2d eltrrels2 dftrrel2 wrel 3bitr4g ) ABCZAADAEZAFCZGQANZGAHCAIPRSQABJKALAMO $. $( Equality theorem for transitive relation. (Contributed by Peter Mazsa, 15-Apr-2019.) (Revised by Peter Mazsa, 23-Sep-2021.) $) trreleq $p |- ( R = S -> ( TrRel R <-> TrRel S ) ) $= ( wceq ccom wrel wa wtrrel coideq id sseq12d releq anbi12d dftrrel2 3bitr4g wss ) ABCZAADZAOZAEZFBBDZBOZBEZFAGBGPRUASUBPQTABABHPIJABKLAMBMN $. $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Equivalence relations =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) $( Define the class of all equivalence relations. The element of the class of all equivalence relations and the equivalence relation predicate are the same when the element is a set, cf. ~ eleqvrelsrel . Alternate definitions are ~ dfeqvrels2 and ~ dfeqvrels3 . (Contributed by Peter Mazsa, 7-Nov-2018.) $) df-eqvrels $a |- EqvRels = ( ( RefRels i^i SymRels ) i^i TrRels ) $. $( Define the equivalence relation predicate. (Read: ` R ` is an equivalence relation.) The element of the class of all equivalence relations and the equivalence relation predicate are the same when ` R ` is a set, cf. ~ eleqvrelsrel . Alternate definitions are ~ dfeqvrel2 and ~ dfeqvrel3 . (Contributed by Peter Mazsa, 17-Apr-2019.) $) df-eqvrel $a |- ( EqvRel R <-> ( RefRel R /\ SymRel R /\ TrRel R ) ) $. $( Define the elementhood equivalence relation predicate. (Read: ` A ` has disjoint elements, or, the elementhood equivalence relation on ` A ` .) We do not use this definition, this is just a placeholder for the sake of the symmetry with ~ df-eldisj . Disjointness of two sets is often defined as no elements in common: we cannot ignore this tradition completely. This is why we need ~ df-eldisj which in our case is a special case of a more general disjointness definition ~ df-disjALTV . We could use a corresponding elementhood equivalence relation (this ~ df-eleqvrel ), which would change ~ mpet3 to a much less intuitive form. Instead, we defined coelements ~ df-coels as a special case of cosets ~ df-coss , and use the general equivalence relation definition with these special cosets, i.e., with coelements, which results in the more intuitive version, ~ mpet3 . (Contributed by Peter Mazsa, 11-Dec-2021.) $) df-eleqvrel $a |- ( ElEqvRel A <-> EqvRel ,~ ( `' _E |` A ) ) $. $( Alternate definition of the class of all equivalence relations. (Contributed by Peter Mazsa, 2-Dec-2019.) $) dfeqvrels2 $p |- EqvRels = { r e. Rels | ( ( _I |` dom r ) C_ r /\ `' r C_ r /\ ( r o. r ) C_ r ) } $= ( ceqvrels cid cv cdm cres wss ccnv wa ccom crels w3a crefrels csymrels cin crab ctrrels df-eqvrels refsymrels2 dftrrels2 ineq12i inrab 3eqtri rabbiia2 df-3an eqtr4i ) BCADZEFUGGZUGHUGGZIZUGUGJUGGZIZAKPZUHUIUKLZAKPBMNOZQOUJAKPZ UKAKPZOUMRUOUPQUQASATUAUJUKAKUBUCUNULAKUHUIUKUEUDUF $. ${ $d r x y z $. $( Alternate definition of the class of all equivalence relations. (Contributed by Peter Mazsa, 2-Dec-2019.) $) dfeqvrels3 $p |- EqvRels = { r e. Rels | ( A. x e. dom r x r x /\ A. x A. y ( x r y -> y r x ) /\ A. x A. y A. z ( ( x r y /\ y r z ) -> x r z ) ) } $= ( cid cv cdm cres wss ccnv ccom w3a wbr wral wi crels ceqvrels dfeqvrels2 wal wa issref cnvsym cotrg 3anbi123i rabbieq ) EDFZGZHUFIZUFJUFIZUFUFKUFI ZLAFZUKUFMAUGNZUKBFZUFMZUMUKUFMOBSASZUNUMCFZUFMTUKUPUFMOCSBSASZLDPQDRUHUL UIUOUJUQAUGUFUAABUFUBABCUFUFUFUCUDUE $. $} $( Alternate definition of the equivalence relation predicate. (Contributed by Peter Mazsa, 22-Apr-2019.) $) dfeqvrel2 $p |- ( EqvRel R <-> ( ( ( _I |` dom R ) C_ R /\ `' R C_ R /\ ( R o. R ) C_ R ) /\ Rel R ) ) $= ( weqvrel wrefrel wsymrel wtrrel w3a cid cdm cres ccnv ccom wrel refsymrel2 wss wa df-eqvrel dftrrel2 anbi12i df-3an anbi1i 3anan32 anandi3r 3bitr2i 3bitr4i bitri ) ABACZADZAEZFZGAHIANZAJANZAAKANZFZALZOZAPUFUGOZUHOUJUKOZUNOZ ULUNOZOZUIUOUPURUHUSAMAQRUFUGUHSUOUQULOZUNOUQUNULFUTUMVAUNUJUKULSTUQUNULUAU QUNULUBUCUDUE $. ${ $d R x y z $. $( Alternate definition of the equivalence relation predicate. (Contributed by Peter Mazsa, 22-Apr-2019.) $) dfeqvrel3 $p |- ( EqvRel R <-> ( ( A. x e. dom R x R x /\ A. x A. y ( x R y -> y R x ) /\ A. x A. y A. z ( ( x R y /\ y R z ) -> x R z ) ) /\ Rel R ) ) $= ( weqvrel wrefrel wsymrel wtrrel w3a cv wbr cdm wral wi wa wrel df-eqvrel wal refsymrel3 df-3an dftrrel3 anbi12i 3anan32 anandi3r 3bitr2i 3bitr4i anbi1i bitri ) DEDFZDGZDHZIZAJZUMDKADLMZUMBJZDKZUOUMDKNBRARZUPUOCJZDKOUMU RDKNCRBRARZIZDPZOZDQUIUJOZUKOUNUQOZVAOZUSVAOZOZULVBVCVEUKVFABDSABCDUAUBUI UJUKTVBVDUSOZVAOVDVAUSIVGUTVHVAUNUQUSTUGVDVAUSUCVDVAUSUDUEUFUH $. $} ${ $d R r $. $( Element of the class of all equivalence relations. (Contributed by Peter Mazsa, 24-Aug-2021.) $) eleqvrels2 $p |- ( R e. EqvRels <-> ( ( ( _I |` dom R ) C_ R /\ `' R C_ R /\ ( R o. R ) C_ R ) /\ R e. Rels ) ) $= ( vr cid cv cdm cres wss ccnv ccom crels ceqvrels dfeqvrels2 wceq reseq2d w3a dmeq id sseq12d cnveqd coeq12d 3anbi123d rabeqel ) CBDZEZFZUCGZUCHZUC GZUCUCIZUCGZOCAEZFZAGZAHZAGZAAIZAGZOBJKABLUCAMZUFUMUHUOUJUQURUEULUCAURUDU KCUCAPNURQZRURUGUNUCAURUCAUSSUSRURUIUPUCAURUCAUCAUSUSTUSRUAUB $. $} ${ $d R r x y z $. $( Element of the class of all equivalence relations. (Contributed by Peter Mazsa, 24-Aug-2021.) $) eleqvrels3 $p |- ( R e. EqvRels <-> ( ( A. x e. dom R x R x /\ A. x A. y ( x R y -> y R x ) /\ A. x A. y A. z ( ( x R y /\ y R z ) -> x R z ) ) /\ R e. Rels ) ) $= ( vr cv wbr cdm wral wi wal wa w3a crels ceqvrels dfeqvrels3 wceq imbi12d breq 2albidv dmeq raleqbidv anbi12d albidv 3anbi123d rabeqel ) AFZUGEFZGZ AUHHZIZUGBFZUHGZULUGUHGZJZBKAKZUMULCFZUHGZLZUGUQUHGZJZCKBKZAKZMUGUGDGZADH ZIZUGULDGZULUGDGZJZBKAKZVGULUQDGZLZUGUQDGZJZCKBKZAKZMENODABCEPUHDQZUKVFUP VJVCVPVQUIVDAUJVEUHDUAUGUGUHDSUBVQUOVIABVQUMVGUNVHUGULUHDSZULUGUHDSRTVQVB VOAVQVAVNBCVQUSVLUTVMVQUMVGURVKVRULUQUHDSUCUGUQUHDSRTUDUEUF $. $} $( The element of the class of all equivalence relations and the equivalence relation predicate are the same when ` R ` is a set. (Contributed by Peter Mazsa, 24-Aug-2021.) $) eleqvrelsrel $p |- ( R e. V -> ( R e. EqvRels <-> EqvRel R ) ) $= ( wcel cid cdm cres wss ccnv ccom w3a crels wrel ceqvrels weqvrel elrelsrel wa anbi2d eleqvrels2 dfeqvrel2 3bitr4g ) ABCZDAEFAGAHAGAAIAGJZAKCZPUBALZPAM CANUAUCUDUBABOQARAST $. $( An equivalence relation is a relation. (Contributed by Peter Mazsa, 2-Jun-2019.) $) eqvrelrel $p |- ( EqvRel R -> Rel R ) $= ( weqvrel cid cdm cres wss ccnv ccom w3a wrel dfeqvrel2 simprbi ) ABCADEAFA GAFAAHAFIAJAKL $. $( An equivalence relation is reflexive. (Contributed by Peter Mazsa, 29-Dec-2021.) $) eqvrelrefrel $p |- ( EqvRel R -> RefRel R ) $= ( weqvrel wrefrel wsymrel wtrrel df-eqvrel simp1bi ) ABACADAEAFG $. $( An equivalence relation is symmetric. (Contributed by Peter Mazsa, 29-Dec-2021.) $) eqvrelsymrel $p |- ( EqvRel R -> SymRel R ) $= ( weqvrel wrefrel wsymrel wtrrel df-eqvrel simp2bi ) ABACADAEAFG $. $( An equivalence relation is transitive. (Contributed by Peter Mazsa, 29-Dec-2021.) $) eqvreltrrel $p |- ( EqvRel R -> TrRel R ) $= ( weqvrel wrefrel wsymrel wtrrel df-eqvrel simp3bi ) ABACADAEAFG $. $( Equivalence relation implies that the domain and the range are equal. (Contributed by Peter Mazsa, 29-Dec-2021.) $) eqvrelim $p |- ( EqvRel R -> dom R = ran R ) $= ( weqvrel wsymrel cdm crn wceq eqvrelsymrel symrelim syl ) ABACADAEFAGAHI $. $( Equality theorem for equivalence relation. (Contributed by Peter Mazsa, 19-Apr-2020.) (Revised by Peter Mazsa, 23-Sep-2021.) $) eqvreleq $p |- ( R = S -> ( EqvRel R <-> EqvRel S ) ) $= ( wceq wrefrel wsymrel wtrrel weqvrel refreleq symreleq 3anbi123d df-eqvrel w3a trreleq 3bitr4g ) ABCZADZAEZAFZLBDZBEZBFZLAGBGOPSQTRUAABHABIABMJAKBKN $. ${ eqvreleqi.1 $e |- R = S $. $( Equality theorem for equivalence relation, inference version. (Contributed by Peter Mazsa, 23-Sep-2021.) $) eqvreleqi $p |- ( EqvRel R <-> EqvRel S ) $= ( wceq weqvrel wb eqvreleq ax-mp ) ABDAEBEFCABGH $. $} ${ eqvreleqd.1 $e |- ( ph -> R = S ) $. $( Equality theorem for equivalence relation, deduction version. (Contributed by Peter Mazsa, 23-Sep-2021.) $) eqvreleqd $p |- ( ph -> ( EqvRel R <-> EqvRel S ) ) $= ( wceq weqvrel wb eqvreleq syl ) ABCEBFCFGDBCHI $. $} ${ eqvrelsym.1 $e |- ( ph -> EqvRel R ) $. eqvrelsym.2 $e |- ( ph -> A R B ) $. $( An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 2-Jun-2019.) $) eqvrelsym $p |- ( ph -> B R A ) $= ( ccnv wbr wrel wb weqvrel eqvrelrel syl relbrcnvg mpbird wsymrel wrefrel wss wtrrel df-eqvrel simp2bi dfsymrel2 simplbi ssbrd mpd ) ACBDGZHZCBDHAU GBCDHZFADIZUGUHJADKZUIEDLMCBDNMOAUFDCBAUJUFDRZEUJDPZUKUJDQULDSDTUAULUKUID UBUCMMUDUE $. $} ${ eqvrelsymb.1 $e |- ( ph -> EqvRel R ) $. $( An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised and distinct variable conditions removed by Peter Mazsa, 2-Jun-2019.) $) eqvrelsymb $p |- ( ph -> ( A R B <-> B R A ) ) $= ( wbr wa weqvrel adantr simpr eqvrelsym impbida ) ABCDFZCBDFZAMGBCDADHZME IAMJKANGCBDAONEIANJKL $. $} ${ $d A x $. $d B x $. $d C x $. $d R x $. eqvreltr.1 $e |- ( ph -> EqvRel R ) $. $( An equivalence relation is transitive. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 2-Jun-2019.) $) eqvreltr $p |- ( ph -> ( ( A R B /\ B R C ) -> A R C ) ) $= ( vx wbr wa ccom cv wex cvv wcel wrel syl simpr brrelex syl2an wss spcegv weqvrel eqvrelrel wceq breq2 breq1 anbi12d sylc wb simpl brrelex2 syl2anc brcog mpbird ex cid cdm cres ccnv w3a dfeqvrel2 simplbi simp3d ssbrd syld ) ABCEHZCDEHZIZBDEEJZHZBDEHAVHVJAVHIZVJBGKZEHZVLDEHZIZGLZVKCMNZVHVPAEOZVG VQVHAEUBZVRFEUCPZVFVGQZCDERSAVHQVOVHGCMVLCUDVMVFVNVGVLCBEUEVLCDEUFUGUAUHV KBMNZDMNZVJVPUIAVRVFWBVHVTVFVGUJBCERSAVRVGWCVHVTWACDEUKSGBDEEMMUMULUNUOAV IEBDAVSVIETZFVSUPEUQURETZEUSETZWDVSWEWFWDUTVREVAVBVCPVDVE $. $} ${ eqvreltrd.1 $e |- ( ph -> EqvRel R ) $. eqvreltrd.2 $e |- ( ph -> A R B ) $. eqvreltrd.3 $e |- ( ph -> B R C ) $. $( A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Peter Mazsa, 2-Jun-2019.) $) eqvreltrd $p |- ( ph -> A R C ) $= ( wbr eqvreltr mp2and ) ABCEICDEIBDEIGHABCDEFJK $. $} ${ eqvreltr4d.1 $e |- ( ph -> EqvRel R ) $. eqvreltr4d.2 $e |- ( ph -> A R B ) $. eqvreltr4d.3 $e |- ( ph -> C R B ) $. $( A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Peter Mazsa, 2-Jun-2019.) $) eqvreltr4d $p |- ( ph -> A R C ) $= ( eqvrelsym eqvreltrd ) ABCDEFGADCEFHIJ $. $} ${ $d A x $. $d R x $. $d ph x $. eqvrelref.1 $e |- ( ph -> EqvRel R ) $. eqvrelref.2 $e |- ( ph -> A e. dom R ) $. $( An equivalence relation is reflexive on its field. Compare Theorem 3M of [Enderton] p. 56. (Contributed by Mario Carneiro, 6-May-2013.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 2-Jun-2019.) $) eqvrelref $p |- ( ph -> A R A ) $= ( vx cv wbr cdm wcel wex weqvrel wrel wb eqvrelrel releldmb 3syl mpbid wa adantr simpr eqvreltr4d exlimddv ) ABFGZCHZBBCHFABCIJZUEFKZEACLZCMUFUGNDC OFBCPQRAUESBUDBCAUHUEDTAUEUAZUIUBUC $. $} ${ $d A x $. $d B x $. $d R x $. $d ph x $. eqvrelth.1 $e |- ( ph -> EqvRel R ) $. eqvrelth.2 $e |- ( ph -> A e. dom R ) $. $( Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.) (Revised by Peter Mazsa, 2-Jun-2019.) $) eqvrelth $p |- ( ph -> ( A R B <-> [ A ] R = [ B ] R ) ) $= ( vx wbr cec wa wcel eqvreltr impl sylan impbida cvv adantr elecg sylancr wb wceq cv simpl eqvrelsymb biimpa jca cdm vex wrel weqvrel eqvrelrel syl brrelex2 3bitr4d eqrdv elecALTV syl2anc mpbird simpr eleqtrd dmec2d mpbid eqvrelref eqvrelsym ) ABCDHZBDIZCDIZUAZAVEJZGVFVGVIBGUBZDHZCVJDHZVJVFKZVJ VGKZVIVKVLVIACBDHZJVKVLVIAVOAVEUCAVEVOABCDEUDUEUFAVOVKVLACBVJDELMNAVEVLVK ABCVJDELMOVIVJPKZBDUGZKZVMVKTGUHZAVRVEFQVJBDPVQRSVIVPCPKZVNVLTVSADUIZVEVT ADUJZWAEDUKULBCDUMNVJCDPPRSUNUOAVHJZCBDAWBVHEQWCBVGKZVOWCBVFVGWCBVFKZBBDH ZAWFVHABDEFVCQWCVRVRWEWFTAVRVHFQZWGBBDVQVQUPUQURAVHUSZUTWCCVQKZVRWDVOTWCV RWIWGWCBCDWHVAVBWGCBDVQVQUPUQVBVDO $. $} ${ eqvrelcl.1 $e |- ( ph -> EqvRel R ) $. eqvrelcl.2 $e |- ( ph -> A R B ) $. $( Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 2-Jun-2019.) $) eqvrelcl $p |- ( ph -> A e. dom R ) $= ( wrel wbr cdm wcel weqvrel eqvrelrel syl releldm syl2anc ) ADGZBCDHBDIJA DKPEDLMFBCDNO $. $} ${ eqvrelthi.1 $e |- ( ph -> EqvRel R ) $. eqvrelthi.2 $e |- ( ph -> A R B ) $. $( Basic property of equivalence relations. Part of Lemma 3N of [Enderton] p. 57. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.) (Revised by Peter Mazsa, 2-Jun-2019.) $) eqvrelthi $p |- ( ph -> [ A ] R = [ B ] R ) $= ( wbr cec wceq eqvrelcl eqvrelth mpbid ) ABCDGBDHCDHIFABCDEABCDEFJKL $. $} ${ $d A x $. $d B x $. $d R x $. $( Equivalence classes do not overlap. In other words, two equivalence classes are either equal or disjoint. Theorem 74 of [Suppes] p. 83. (Contributed by NM, 15-Jun-2004.) (Revised by Mario Carneiro, 9-Jul-2014.) (Revised by Peter Mazsa, 2-Jun-2019.) $) eqvreldisj $p |- ( EqvRel R -> ( [ A ] R = [ B ] R \/ ( [ A ] R i^i [ B ] R ) = (/) ) ) $= ( vx weqvrel cec cin c0 wceq wn wcel wbr adantl cvv wb ecexr syl elecALTV sylancl mpbid cv wex neq0 simpl elin simplbi simprbi eqvreltr4d eqvrelthi wa vex ex exlimdv syl5bi orrd orcomd ) CEZACFZBCFZGZHIZURUSIZUQVAVBVAJDUA ZUTKZDUBUQVBDUTUCUQVDVBDUQVDVBUQVDUJZABCUQVDUDZVEAVCBCVFVEVCURKZAVCCLZVDV GUQVDVGVCUSKZVCURUSUEZUFMZVEANKZVCNKZVGVHOVEVGVLVKVCACPQDUKZAVCCNNRSTVEVI BVCCLZVDVIUQVDVGVIVJUGMZVEBNKZVMVIVOOVEVIVQVPVCBCPQVNBVCCNNRSTUHUIULUMUNU OUP $. $} ${ $d A x y $. $d B x $. $d C x y $. $d R x y $. $d ph x y $. qsdisjALTV.1 $e |- ( ph -> EqvRel R ) $. qsdisjALTV.2 $e |- ( ph -> B e. ( A /. R ) ) $. qsdisjALTV.3 $e |- ( ph -> C e. ( A /. R ) ) $. $( Elements of a quotient set do not overlap. (Contributed by Rodolfo Medina, 12-Oct-2010.) (Revised by Mario Carneiro, 11-Jul-2014.) (Revised by Peter Mazsa, 3-Jun-2019.) $) qsdisjALTV $p |- ( ph -> ( B = C \/ ( B i^i C ) = (/) ) ) $= ( vx vy wcel wceq cin c0 wo cv cec eqeq1d orbi12d wa cqs eqid eqeq1 ineq1 adantr eqeq2 ineq2 weqvrel ad2antrr eqvreldisj syl ectocld mpdan ) ACBEUA ZKCDLZCDMZNLZOZGIPZEQZDLZUTDMZNLZOZURAICBEUNUNUBZUTCLZVAUOVCUQUTCDUCVFVBU PNUTCDUDRSAUSBKZTZDUNKZVDAVIVGHUEUTJPZEQZLZUTVKMZNLZOZVDVHJDBEUNVEVKDLZVL VAVNVCVKDUTUFVPVMVBNVKDUTUGRSVHVJBKZTEUHZVOAVRVGVQFUIUSVJEUJUKULUMULUM $. $} $( Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 4-Jul-2020.) (Revised by Peter Mazsa, 20-Dec-2021.) $) eqvrelcoss $p |- ( EqvRel ,~ R <-> TrRel ,~ R ) $= ( weqvrel wrefrel wsymrel wtrrel w3a df-eqvrel refrelcoss symrelcoss bitr4i ccoss triantru3 ) AKZBMCZMDZMEZFPMGNOPAHAILJ $. ${ $d R x y z $. $( Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 28-Apr-2019.) $) eqvrelcoss3 $p |- ( EqvRel ,~ R <-> A. x A. y A. z ( ( x ,~ R y /\ y ,~ R z ) -> x ,~ R z ) ) $= ( cv ccoss wbr cdm wral wi wal wrel weqvrel relcoss biantru refrelcosslem wa w3a symrelcoss3 simpli triantru3 dfeqvrel3 3bitr4ri ) AEZUDDFZGAUEHIZU DBEZUEGZUGUDUEGJBKAKZUHUGCEZUEGQUDUJUEGJCKBKAKZRZULUELZQUKUEMUMULDNOUFUIU KADPUIUMABDSTUAABCUEUBUC $. $} ${ $d R x y z $. $( Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 3-May-2019.) $) eqvrelcoss2 $p |- ( EqvRel ,~ R <-> ,~ ,~ R C_ ,~ R ) $= ( vx vy vz ccoss weqvrel cv wbr wa wi wal wss eqvrelcoss3 cocossss bitr4i ) AEZFBGZCGZPHRDGZPHIQSPHJDKCKBKPEPLBCDAMBCDAPNO $. $} ${ $d R x y z $. $( Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 3-May-2019.) (Revised by Peter Mazsa, 30-Sep-2021.) $) eqvrelcoss4 $p |- ( EqvRel ,~ R <-> A. x A. z ( ( [ x ] ,~ R i^i [ z ] ,~ R ) =/= (/) -> ( [ x ] `' R i^i [ z ] `' R ) =/= (/) ) ) $= ( vy ccoss weqvrel cv wbr wa wi wal cec cin c0 wne ccnv eqvrelcoss3 bitri trcoss2 ) CEZFAGZDGZTHUBBGZTHIUAUCTHJBKDKAKUATLUCTLMNOUACPZLUCUDLMNOJBKAK ADBCQADBCSR $. $} ${ $d A x z $. $( Two ways to express equivalent coelements. (Contributed by Peter Mazsa, 20-Oct-2021.) $) eqvrelcoels4 $p |- ( EqvRel ~ A <-> A. x A. z ( ( [ x ] ~ A i^i [ z ] ~ A ) =/= (/) -> ( [ x ] `' ( `' _E |` A ) i^i [ z ] `' ( `' _E |` A ) ) =/= (/) ) ) $= ( cep ccnv cres ccoss weqvrel cv cec cin c0 wne wi wal ccoels eqvrelcoss4 df-coels eqvreleqi eceq2i ineq12i neeq1i imbi1i 2albii 3bitr4i ) DECFZGZH AIZUGJZBIZUGJZKZLMZUHUFEZJUJUNJKLMZNZBOAOCPZHUHUQJZUJUQJZKZLMZUONZBOAOABU FQUQUGCRZSVBUPABVAUMUOUTULLURUIUSUKUQUGUHVCTUQUGUJVCTUAUBUCUDUE $. $} ${ $d A x $. $d B x $. $d C x $. $d R x $. $( If an element of a quotient set contains a given element, it is equal to the equivalence class of the element. (Contributed by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 28-Dec-2019.) $) eqvrelqsel $p |- ( ( EqvRel R /\ B e. ( A /. R ) /\ C e. B ) -> B = [ C ] R ) $= ( vx weqvrel cqs wcel cec wceq cv wi eqid eleq2 eqeq1 imbi12d wbr wa cvv wb elecALTV el2v1 ibi simpll simpr eqvrelthi ex syl5 ectocld 3impia ) DFZ BADGZHCBHZBCDIZJZCEKZDIZHZUQUNJZLUMUOLUKEBADULULMUQBJURUMUSUOUQBCNUQBUNOP URUPCDQZUKUPAHZRZUSURUTURURUTTEUPCDSUQUAUBUCVBUTUSVBUTRUPCDUKVAUTUDVBUTUE UFUGUHUIUJ $. $} $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Domain quotients =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) ${ $d x y $. $( Define the class of all domain quotients. Domain quotients are pairs of sets, typically a relation and a set, where the quotient (cf. ~ df-qs ) of the relation on its domain is equal to the set. See comments of ~ df-ers for the motivation for this definition. (Contributed by Peter Mazsa, 16-Apr-2019.) $) df-dmqss $a |- DomainQss = { <. x , y >. | ( dom x /. x ) = y } $. $} $( Define the domain quotient predicate. (Read: the domain quotient of ` R ` is ` A ` .) If ` A ` and ` R ` are sets, the domain quotient binary relation and the domain quotient predicate are the same, cf. ~ brdmqssqs . (Contributed by Peter Mazsa, 9-Aug-2021.) $) df-dmqs $a |- ( R DomainQs A <-> ( dom R /. R ) = A ) $. $( Equality theorem for domain quotient. (Contributed by Peter Mazsa, 17-Apr-2019.) $) dmqseq $p |- ( R = S -> ( dom R /. R ) = ( dom S /. S ) ) $= ( wceq cdm cqs dmeq qseq12 sylan anidms ) ABCZADZAEBDZBECZJKLCJMABFKLABGHI $. ${ dmqseqi.1 $e |- R = S $. $( Equality theorem for domain quotient, inference version. (Contributed by Peter Mazsa, 26-Sep-2021.) $) dmqseqi $p |- ( dom R /. R ) = ( dom S /. S ) $= ( wceq cdm cqs dmqseq ax-mp ) ABDAEAFBEBFDCABGH $. $} ${ dmqseqd.1 $e |- ( ph -> R = S ) $. $( Equality theorem for domain quotient set, deduction version. (Contributed by Peter Mazsa, 23-Apr-2021.) $) dmqseqd $p |- ( ph -> ( dom R /. R ) = ( dom S /. S ) ) $= ( wceq cdm cqs dmqseq syl ) ABCEBFBGCFCGEDBCHI $. $} $( Equality theorem for domain quotient. (Contributed by Peter Mazsa, 17-Apr-2019.) $) dmqseqeq1 $p |- ( R = S -> ( ( dom R /. R ) = A <-> ( dom S /. S ) = A ) ) $= ( wceq cdm cqs dmqseq eqeq1d ) BCDBEBFCECFABCGH $. ${ dmqseqeq1i.1 $e |- R = S $. $( Equality theorem for domain quotient, inference version. (Contributed by Peter Mazsa, 26-Sep-2021.) $) dmqseqeq1i $p |- ( ( dom R /. R ) = A <-> ( dom S /. S ) = A ) $= ( wceq cdm cqs wb dmqseqeq1 ax-mp ) BCEBFBGAECFCGAEHDABCIJ $. $} ${ dmqseqeq1d.1 $e |- ( ph -> R = S ) $. $( Equality theorem for domain quotient set, deduction version. (Contributed by Peter Mazsa, 26-Sep-2021.) $) dmqseqeq1d $p |- ( ph -> ( ( dom R /. R ) = A <-> ( dom S /. S ) = A ) ) $= ( wceq cdm cqs wb dmqseqeq1 syl ) ACDFCGCHBFDGDHBFIEBCDJK $. $} ${ $d A x y $. $d R x y $. $( The domain quotient binary relation. (Contributed by Peter Mazsa, 17-Apr-2019.) $) brdmqss $p |- ( ( A e. V /\ R e. W ) -> ( R DomainQss A <-> ( dom R /. R ) = A ) ) $= ( vx vy wcel cdmqss wbr cdm cqs wb dmqseq eqeqan1d df-dmqss brabga ancoms wceq cv ) BDGACGBAHIBJBKZARZLESZJUBKZFSZRUAEFBAHDCUBBRUCTUDAUBBMNEFOPQ $. $} $( If ` A ` and ` R ` are sets, the domain quotient binary relation and the domain quotient predicate are the same. (Contributed by Peter Mazsa, 14-Aug-2021.) $) brdmqssqs $p |- ( ( A e. V /\ R e. W ) -> ( R DomainQss A <-> R DomainQs A ) ) $= ( wcel wa cdmqss wbr cdm cqs wceq wdmqs brdmqss df-dmqs syl6bbr ) ACEBDEFBA GHBIBJAKABLABCDMABNO $. $( The empty set is not an element of a domain quotient. (Contributed by Peter Mazsa, 2-Mar-2018.) $) n0eldmqs $p |- -. (/) e. ( dom R /. R ) $= ( c0 cdm cqs wcel wn wss ssid n0elqs mpbir ) BACZADEFKKGKHKAIJ $. $( The empty set is not an element of a domain quotient. (Contributed by Peter Mazsa, 3-Nov-2018.) $) n0eldmqseq $p |- ( ( dom R /. R ) = A -> -. (/) e. A ) $= ( cdm cqs wceq c0 wcel wn n0eldmqs eleq2 notbid mpbii ) BCBDZAEZFMGZHFAGZHB INOPMAFJKL $. $( Implication of the equality of a domain quotient set and a quotient set. (Contributed by Peter Mazsa, 16-Sep-2020.) $) dmqseqqsim $p |- ( ( dom S /. S ) = ( A /. R ) -> dom ( R |` A ) = A ) $= ( cdm cqs wceq c0 wcel wn cres n0eldmqs eleq2 notbid mpbii n0elqs2 sylib ) CDCEZABEZFZGRHZIZBAJDAFSGQHZIUACKSUBTQRGLMNABOP $. ${ $d A x y $. $( Two ways of expressing that the empty set is not an element of a class. (Contributed by Peter Mazsa, 27-May-2021.) $) n0el3 $p |- ( -. (/) e. A <-> ( dom ( `' _E |` A ) /. ( `' _E |` A ) ) = A ) $= ( vx vy c0 wcel wn cep ccnv cres cdm cqs wceq cv wa wex wral n0el dmopab3 copab bitri cnvepres dmeqi eqeq1i bitr4i biimpi qseq1d qsresid qsid eqtri syl6eq eqeq2i dmqseqqsim sylbir n0el2 sylibr impbii ) DAEFZGHZAIZJZUSKZAL ZUQVAAUSKZAUQUTAUSUQUTALZUQBMZAECMVEEZNBCSZJZALZVDUQVFCOBAPVIBCAQVFBCARTU TVHAUSVGBCAUAUBUCUDUEUFVCAURKZAAURUGAUHZUIUJVBVDUQVBVAVJLVDVJAVAVKUKAURUS ULUMAUNUOUP $. $} $( The domain quotient binary relation of the restricted converse epsilon relation is equivalent to the negated elementhood of the empty set in the restriction. (Contributed by Peter Mazsa, 14-Aug-2021.) $) cnvepresdmqss $p |- ( A e. V -> ( ( `' _E |` A ) DomainQss A <-> -. (/) e. A ) ) $= ( wcel cep ccnv cres cdmqss wbr cdm cqs wceq c0 wn cnvepresex brdmqss mpdan cvv wb n0el3 syl6bbr ) ABCZDEAFZAGHZUBIUBJAKZLACMUAUBQCUCUDRABNAUBBQOPAST $. $( The domain quotient predicate for the restricted converse epsilon relation is equivalent to the negated elementhood of the empty set in the restriction. (Contributed by Peter Mazsa, 14-Aug-2021.) $) cnvepresdmqs $p |- ( ( `' _E |` A ) DomainQs A <-> -. (/) e. A ) $= ( cep ccnv cres wdmqs cdm cqs wceq c0 wcel wn df-dmqs n0el3 bitr4i ) ABCADZ EOFOGAHIAJKAOLAMN $. $( The range of a relation is equal to the union of the domain quotient. (Contributed by Peter Mazsa, 13-Oct-2018.) $) unidmqs $p |- ( R e. V -> ( Rel R -> U. ( dom R /. R ) = ran R ) ) $= ( wcel wrel cdm cqs cuni crn wceq wa cres cvv resexg rnresequniqs syl resdm adantr rneqd adantl eqtr3d ex ) ABCZADZAEZAFGZAHZIUBUCJAUDKZHZUEUFUBUHUEIZU CUBUGLCUIAUDBMUDALNOQUCUHUFIUBUCUGAAPRSTUA $. $( The union of the domain quotient of a relation is equal to the class ` A ` if and only if the range is equal to it as well. (Contributed by Peter Mazsa, 21-Apr-2019.) (Revised by Peter Mazsa, 28-Dec-2021.) $) unidmqseq $p |- ( R e. V -> ( Rel R -> ( U. ( dom R /. R ) = A <-> ran R = A ) ) ) $= ( wcel wrel cdm cqs cuni wceq crn wb wa unidmqs imp eqeq1d ex ) BCDZBEZBFBG HZAIBJZAIKQRLSTAQRSTIBCMNOP $. $( If the domain quotient of a relation is equal to the class ` A ` , then the range of the relation is the union of the class. (Contributed by Peter Mazsa, 29-Dec-2021.) $) dmqseqim $p |- ( R e. V -> ( Rel R -> ( ( dom R /. R ) = A -> ran R = U. A ) ) ) $= ( wcel wrel cdm cqs wceq crn cuni wi wa unieq a1i wb unidmqseq imp sylibd ex ) BCDZBEZBFBGZAHZBIAJZHZKTUALZUCUBJUDHZUEUCUGKUFUBAMNTUAUGUEOUDBCPQRS $. $( Lemma for ~ erALTimlem2 . (Contributed by Peter Mazsa, 29-Dec-2021.) $) dmqseqim2 $p |- ( R e. V -> ( Rel R -> ( ( dom R /. R ) = A -> ( B e. ran R <-> B e. U. A ) ) ) ) $= ( wcel wrel cdm cqs wceq crn cuni wb dmqseqim eleq2 syl8 ) CDECFCGCHAICJZAK ZIBPEBQELACDMPQBNO $. ${ $d A u x $. $d R u x $. $( Elementhood in the domain quotient of a relation. (Contributed by Peter Mazsa, 24-Apr-2021.) $) releldmqs $p |- ( A e. V -> ( Rel R -> ( A e. ( dom R /. R ) <-> E. u e. dom R E. x e. [ u ] R A = [ u ] R ) ) ) $= ( wcel wrel cdm cqs cv cec wceq wrex wb cres resdm dmqseqd eleq2d adantl wa eldmqsres2 adantr bitr3d ex ) CEFZDGZCDHZDIZFZCBJDKZLAUJMBUGMZNUEUFTCD UGOZHULIZFZUIUKUFUNUINUEUFUMUHCUFULDDPQRSUEUNUKNUFABUGCDEUAUBUCUD $. $} ${ $d A u x $. $d B u x $. $d R u x $. $( Elementhood in the domain quotient of the class of cosets by restriction. (Contributed by Peter Mazsa, 4-May-2019.) $) eldmqs1cossres $p |- ( B e. V -> ( B e. ( dom ,~ ( R |` A ) /. ,~ ( R |` A ) ) <-> E. u e. A E. x e. [ u ] R B = [ x ] ,~ ( R |` A ) ) ) $= ( wcel cres ccoss cdm cqs cv cec wrex wceq wa wex df-rex exbii bitri cvv elqsg wb eldm1cossres2 elv anbi1i syl6bb rexbii rexcom4 r19.41v syl6bbr ) DFGZDECHIZJZUMKGZALZBLEMZGZBCNZDUPUMMOZPZAQZUTAUQNZBCNZULUOUTAUNNZVBAUNDU MFUBVEUPUNGZUTPZAQVBUTAUNRVGVAAVFUSUTVFUSUCABCUPEUAUDUEUFSTUGVDURUTPZAQZB CNZVBVCVIBCUTAUQRUHVJVHBCNZAQVBVHBACUIVKVAAURUTBCUJSTTUK $. $} ${ $d A u x $. $d R u x $. $( Elementhood in the domain quotient of the class of cosets by relation. (Contributed by Peter Mazsa, 23-Apr-2021.) $) releldmqscoss $p |- ( A e. V -> ( Rel R -> ( A e. ( dom ,~ R /. ,~ R ) <-> E. u e. dom R E. x e. [ u ] R A = [ x ] ,~ R ) ) ) $= ( wcel wrel ccoss cdm cqs cv cec wceq wrex wb cres eldmqs1cossres rexbidv wa adantl adantr resdm cosseqd dmqseqd eleq2d eceq2d eqeq2d 3bitr3d ex ) CEFZDGZCDHZIULJZFZCAKZULLZMZABKDLZNZBDIZNZOUJUKSCDUTPZHZIVCJZFZCUOVCLZMZA URNZBUTNZUNVAUJVEVIOUKABUTCDEQUAUKVEUNOUJUKVDUMCUKVCULUKVBDDUBUCZUDUETUKV IVAOUJUKVHUSBUTUKVGUQAURUKVFUPCUKVCULUOVJUFUGRRTUHUI $. $} $( Two ways to express the equality of the domain quotient of the coelements on the class ` A ` with the class ` A ` . (Contributed by Peter Mazsa, 26-Sep-2021.) $) dmqscoelseq $p |- ( ( dom ~ A /. ~ A ) = A <-> ( U. A /. ~ A ) = A ) $= ( ccoels cdm cqs cuni dmcoels qseq1i eqeq1i ) ABZCZIDAEZIDAJKIAFGH $. $( Two ways to express the equality of the domain quotient of the coelements on the class ` A ` with the class ` A ` . (Contributed by Peter Mazsa, 26-Sep-2021.) $) dmqs1cosscnvepreseq $p |- ( ( dom ,~ ( `' _E |` A ) /. ,~ ( `' _E |` A ) ) = A <-> ( U. A /. ~ A ) = A ) $= ( cep ccnv cres ccoss cdm cqs ccoels df-coels dmqseqeq1i dmqscoelseq bitr3i wceq cuni ) BCADEZFOGAMAHZFPGAMANPGAMAPOAIJAKL $. $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Equivalence relations on domain quotients =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) $( Define the class of all equivalence relations on domain quotients (or: domain quotients restricted to equivalence relations). The present definition of equivalence relation in set.mm ~ df-er "is not standard", "somewhat cryptic", has no costant 0-ary class and does not follow the traditional transparent reflexive-symmetric-transitive relation way of definition of equivalence. The definitions ~ df-eqvrels , ~ dfeqvrels2 , ~ dfeqvrels3 and ~ df-eqvrel , ~ dfeqvrel2 , ~ dfeqvrel3 are fully transparent in this regard. However, they lack the domain component ( ` dom R = A ` ) of the present ~ df-er . While we acknowledge the need of a domain component, the present ~ df-er definition does not utilize the results revealed by the new theorems in the Partition-Equivalence Theorem part below (like ~ pets and ~ pet ). From those theorems follows that the natural domain of equivalence relations is not ` R Domain A ` (i.e. ` dom R = A ` cf. ~ brdomaing ), but ` R DomainQss A ` (i.e. ` ( dom R /. R ) = A ` , cf. ~ brdmqss ), cf. ~ erALTimlem vs. ~ prter3 . While I'm sure we need both equivalence relation ~ df-eqvrels and equivalence relation on domain quotient ~ df-ers , I'm not sure whether we need a third equivalence relation concept with the present ` dom R = A ` component as well: this needs further investigation. As a default I suppose that these two concepts ~ df-eqvrels and ~ df-ers are enough and named the predicate version of the one on domain quotient as the alternate version ~ df-erALTV of the present ~ df-er . (Contributed by Peter Mazsa, 26-Jun-2021.) $) df-ers $a |- Ers = ( DomainQss |` EqvRels ) $. $( Equivalence relation with natural domain predicate, cf. the comment of ~ df-ers . Alternate definition is ~ dferALT2 . Binary equivalence relation with natural domain and the equivalence relation with natural domain predicate are the same when ` A ` and ` R ` are sets, cf. ~ brerserALTV . (Contributed by Peter Mazsa, 12-Aug-2021.) $) df-erALTV $a |- ( R ErALTV A <-> ( EqvRel R /\ R DomainQs A ) ) $. $( Define the membership equivalence relation on the class ` A ` (or, the restricted elementhood equivalence relation on its domain quotient ` A ` .) Alternate definitions are ~ dfmember2 and ~ dfmember3 . Later on, in an application of set theory I make a distinction between the default elementhood concept and a special membership concept: membership equivalence relation will be an integral part of that membership concept. (Contributed by Peter Mazsa, 26-Jun-2021.) (Revised by Peter Mazsa, 25-Sep-2021.) $) df-member $a |- ( MembEr A <-> ~ A ErALTV A ) $. $( Binary equivalence relation with natural domain, cf. the comment of ~ df-ers . (Contributed by Peter Mazsa, 23-Jul-2021.) $) brers $p |- ( A e. V -> ( R Ers A <-> ( R e. EqvRels /\ R DomainQss A ) ) ) $= ( ceqvrels cers cdmqss df-ers eqres ) BADEFCGH $. $( Equivalence relation with natural domain predicate, cf. the comment of ~ df-ers . (Contributed by Peter Mazsa, 26-Jun-2021.) (Revised by Peter Mazsa, 30-Aug-2021.) $) dferALT2 $p |- ( R ErALTV A <-> ( EqvRel R /\ ( dom R /. R ) = A ) ) $= ( werALTV weqvrel wdmqs wa cdm cqs wceq df-erALTV df-dmqs anbi2i bitri ) AB CBDZABEZFNBGBHAIZFABJOPNABKLM $. $( Equality theorem for equivalence relation on domain quotient. (Contributed by Peter Mazsa, 25-Sep-2021.) $) erALTeq1 $p |- ( R = S -> ( R ErALTV A <-> S ErALTV A ) ) $= ( wceq weqvrel cdm cqs werALTV eqvreleq dmqseqeq1 anbi12d dferALT2 3bitr4g wa ) BCDZBEZBFBGADZNCEZCFCGADZNABHACHOPRQSBCIABCJKABLACLM $. ${ erALTeq1i.1 $e |- R = S $. $( Equality theorem for equivalence relation on domain quotient, inference version. (Contributed by Peter Mazsa, 25-Sep-2021.) $) erALTeq1i $p |- ( R ErALTV A <-> S ErALTV A ) $= ( wceq werALTV wb erALTeq1 ax-mp ) BCEABFACFGDABCHI $. $} ${ erALTeq1d.1 $e |- ( ph -> R = S ) $. $( Equality theorem for equivalence relation on domain quotient, deduction version. (Contributed by Peter Mazsa, 25-Sep-2021.) $) erALTeq1d $p |- ( ph -> ( R ErALTV A <-> S ErALTV A ) ) $= ( wceq werALTV wb erALTeq1 syl ) ACDFBCGBDGHEBCDIJ $. $} $( Alternate definition of the membership equivalence relation. (Contributed by Peter Mazsa, 25-Sep-2021.) $) dfmember2 $p |- ( MembEr A <-> ,~ ( `' _E |` A ) ErALTV A ) $= ( wmember ccoels werALTV ccnv cres ccoss df-member df-coels erALTeq1i bitri cep ) ABAACZDALEAFGZDAHAMNAIJK $. $( Alternate definition of the membership equivalence relation. (Contributed by Peter Mazsa, 26-Sep-2021.) $) dfmember3 $p |- ( MembEr A <-> ( EqvRel ~ A /\ ( U. A /. ~ A ) = A ) ) $= ( wmember ccoels werALTV weqvrel cdm cqs wceq wa cuni df-member dmqscoelseq dferALT2 anbi2i 3bitri ) ABAACZDPEZPFPGAHZIQAJPGAHZIAKAPMRSQALNO $. $( Two ways to express membership equivalence relation on its domain quotient. (Contributed by Peter Mazsa, 26-Sep-2021.) $) eqvreldmqscoels $p |- ( ( EqvRel ,~ ( `' _E |` A ) /\ ( dom ,~ ( `' _E |` A ) /. ,~ ( `' _E |` A ) ) = A ) <-> ( EqvRel ~ A /\ ( U. A /. ~ A ) = A ) ) $= ( cep ccnv cres ccoss weqvrel ccoels cdm cqs wceq df-coels eqvreleqi bicomi cuni dmqs1cosscnvepreseq anbi12i ) BCADEZFZAGZFZQHQIAJANSIAJTRSQAKLMAOP $. $( Binary equivalence relation with natural domain and the equivalence relation with natural domain predicate are the same when ` A ` and ` R ` are sets. (Contributed by Peter Mazsa, 25-Aug-2021.) $) brerserALTV $p |- ( ( A e. V /\ R e. W ) -> ( R Ers A <-> R ErALTV A ) ) $= ( wcel wa cers ceqvrels cdmqss werALTV wb brers adantr weqvrel eleqvrelsrel wbr wdmqs adantl brdmqssqs anbi12d df-erALTV syl6bbr bitrd ) ACEZBDEZFZBAGP ZBHEZBAIPZFZABJZUDUGUJKUEABCLMUFUJBNZABQZFUKUFUHULUIUMUEUHULKUDBDORABCDSTAB UAUBUC $. ${ $d A u x y $. $d R u x y $. $d V x y $. $( Equivalence relation on its natural domain implies that the class of coelements on the domain is equal to the relation (former ~ prter3 in a more convenient form, cf. ~ erALTimlem ). (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 29-Dec-2021.) $) erALTimlem2 $p |- ( R e. V -> ( ( EqvRel R /\ ( dom R /. R ) = A ) -> ~ A = R ) ) $= ( vx vy vu wcel wceq wa wrel anass wbr wrex cvv el2v simpll eleq2d adantl cv wb weqvrel cdm ccoels eqvrelrel pm4.71ri anbi1i bitri relcoels brcoels cqs cec simprl simplr eleqtrrd simprr eqvrelqsel syl3anc elecALTV anassrs bibi2i sylib pm5.32da rexbidva simpr eqvrelcl adantll crn eqvrelim adantr cuni anbi2i imp31 adantrl sylbir sylbi bitrd eluni2 syl6bb mpbid pm4.71rd dmqseqim2 ex r19.41v syl6bbr bitr4d syl5bb eqbrrdv2 mpanl1 an12s sylan2b ) BCGZBUAZBUBZBUJZAHZIZAUCZBHZWPWKBJZWPIZWRWPWSWLIZWOIWTWLXAWOWLWSBUDUEUF WSWLWOKUGZWSWKWPWRWQJZWSWKWPIZWRAUHXDDEWQBXDDSZESZWQLZXEXFBLZTXCWSIXGXEFS ZGZXFXIGZIZFAMZXDXHXGXMTDEFAXEXFNNUIOXDXMXJXHIZFAMZXHWPXMXOTWKWPXLXNFAWPX IAGZIXJXKXHWPXPXJXKXHTZWPXPXJIZIZXKXFXEBUKZGZTXQXSXIXTXFXSWLXIWNGXJXIXTHW LWOXRPXSXIAWNWPXPXJULWLWOXRUMUNWPXPXJUOWMXIXEBUPUQQYAXHXKYAXHTDEXEXFBNNUR OUTVAUSVBVCRXDXHXJFAMZXHIXOXDXHYBXDXHYBXDXHIXEWMGZYBWPXHYCWKWPXHIXEXFBWLW OXHPWPXHVDVEVFXDYCYBTXHXDYCXEAVJGZYBXDYCXEBVGZGZYDXDWMYEXEWPWMYEHZWKWLYGW OBVHVIRQXDWKWTIZYFYDTZWPWTWKXBVKYHWKWSIZWPIYIWKWSWPKYJWOYIWLWKWSWOYIAXEBC WAVLVMVNVOVPFXEAVQVRVIVSWBVTXJXHFAWCWDWEWFRWGWHWIWJWB $. $} $( Equivalence relation on its natural domain implies that the class of coelements on the domain is equal to the relation (former ~ prter3 in the most convenient form, cf. ~ erALTimlem2 ). If you succeed to turn this into petlem.1 given ~ eqvrelqseqdisj2 , you would get a more general form ` ( R Part A <-> ,~ R ErALTV A ) ` of the "pet" theorems like ~ mpet or ~ pet , cf. the comment of ~ pet . (Contributed by Peter Mazsa, 7-Oct-2021.) (Revised by Peter Mazsa, 29-Dec-2021.) $) erALTimlem $p |- ( R e. V -> ( R ErALTV A -> ~ A = R ) ) $= ( werALTV weqvrel cdm cqs wceq wa wcel ccoels dferALT2 erALTimlem2 syl5bi ) ABDBEBFBGAHIBCJAKBHABLABCMN $. $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Functions =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) $( Define the class of all function sets (but not necessarily function relations, cf. ~ df-funsALTV ). It is used only by ~ df-funsALTV . (Contributed by Peter Mazsa, 17-Jul-2021.) $) df-funss $a |- Funss = { x | ,~ x e. CnvRefRels } $. $( Define the function relations class, i.e., the class of all functions. Alternate definitions are ~ dffunsALTV , ... , ~ dffunsALTV5 . (Contributed by Peter Mazsa, 17-Jul-2021.) $) df-funsALTV $a |- FunsALTV = ( Funss i^i Rels ) $. $( Define the function relation predicate, i.e., the function predicate. This definition of the function predicate (based on a more general, converse reflexive, relation) and the original definition of function in set.mm ~ df-fun , are always the same, that is ` ( FunALTV F <-> Fun F ) ` , cf. ~ funALTVfun . The element of the class of all functions and the function predicate are the same, that is ` ( F e. FunsALTV <-> FunALTV F ) ` when ` F ` is a set, cf. ~ elfunsALTVfunALTV . Alternate definitions are ~ dffunALTV2 , ... , ~ dffunALTV5 . (Contributed by Peter Mazsa, 17-Jul-2021.) $) df-funALTV $a |- ( FunALTV F <-> ( CnvRefRel ,~ F /\ Rel F ) ) $. $( Alternate definition of the class of all functions. (Contributed by Peter Mazsa, 18-Jul-2021.) $) dffunsALTV $p |- FunsALTV = { f e. Rels | ,~ f e. CnvRefRels } $= ( ccoss ccnvrefrels wcel cfunsALTV cfunss crels df-funsALTV df-funss abeqin cv ) AKBCDAEFGHAIJ $. $( Alternate definition of the class of all functions. (Contributed by Peter Mazsa, 30-Aug-2021.) $) dffunsALTV2 $p |- FunsALTV = { f e. Rels | ,~ f C_ _I } $= ( ccoss ccnvrefrels wcel cid crels cfunsALTV dffunsALTV cosselrels biantrud cv wss wa cosselcnvrefrels2 syl6rbbr rabimbieq ) AKZBZCDZRELZAFGAHQFDZTTRFD ZMSUAUBTQFIJQNOP $. ${ $d f u x y $. $( Alternate definition of the class of all functions. For the ` X ` axis and the ` Y ` axis you can convert the right side to { f e. Rels | A. x1 A. y1 A. y2 ( ( x1 f y1 /\ x1 f y2 ) -> y1 = y2 ) }. (Contributed by Peter Mazsa, 30-Aug-2021.) $) dffunsALTV3 $p |- FunsALTV = { f e. Rels | A. u A. x A. y ( ( u f x /\ u f y ) -> x = y ) } $= ( cv ccoss ccnvrefrels wcel wbr wa wceq wi wal crels cfunsALTV dffunsALTV cosselrels biantrud cosselcnvrefrels3 syl6rbbr rabimbieq ) DEZFZGHZCEZAEZ UBIUEBEZUBIJUFUGKLBMAMCMZDNODPUBNHZUHUHUCNHZJUDUIUJUHUBNQRABCUBSTUA $. $} ${ $d f u x $. $( Alternate definition of the class of all functions. For the ` X ` axis and the ` Y ` axis you can convert the right side to ` { f e. Rels | A. x1 E* y1 x1 f y1 } ` . (Contributed by Peter Mazsa, 31-Aug-2021.) $) dffunsALTV4 $p |- FunsALTV = { f e. Rels | A. u E* x u f x } $= ( cv ccoss ccnvrefrels wcel wbr wmo crels cfunsALTV dffunsALTV cosselrels wal wa biantrud cosselcnvrefrels4 syl6rbbr rabimbieq ) CDZEZFGZBDADTHAIBN ZCJKCLTJGZUCUCUAJGZOUBUDUEUCTJMPABTQRS $. $} ${ $d f u x y $. $( Alternate definition of the class of all functions. (Contributed by Peter Mazsa, 31-Aug-2021.) $) dffunsALTV5 $p |- FunsALTV = { f e. Rels | A. x e. ran f A. y e. ran f ( x = y \/ ( [ x ] `' f i^i [ y ] `' f ) = (/) ) } $= ( vu cfunsALTV cv wbr wmo wal crels crab wceq ccnv cec cin c0 wo crn wral dffunsALTV4 ineccnvmo2 rabbiia2 eqtr4i ) EDFAFZCFZGAHDIZCJKUDBFZLUDUEMZNU GUHNOPLQBUERZSAUISZCJKADCTUJUFCJABDUEUAUBUC $. $} $( Alternate definition of the function relation predicate, cf. ~ dfdisjALTV2 . (Contributed by Peter Mazsa, 8-Feb-2018.) $) dffunALTV2 $p |- ( FunALTV F <-> ( ,~ F C_ _I /\ Rel F ) ) $= ( wfunALTV ccoss wcnvrefrel wrel wa cid wss df-funALTV cnvrefrelcoss2 bitri anbi1i ) ABACZDZAEZFMGHZOFAINPOAJLK $. ${ $d F u x y $. $( Alternate definition of the function relation predicate, cf. ~ dfdisjALTV3 . Reproduction of ~ dffun2 . For the ` X ` axis and the ` Y ` axis you can convert the right side to ( A. x1 A. y1 A. y2 ( ( x1 f y1 /\ x1 f y2 ) -> y1 = y2 ) /\ Rel F ). (Contributed by NM, 29-Dec-1996.) $) dffunALTV3 $p |- ( FunALTV F <-> ( A. u A. x A. y ( ( u F x /\ u F y ) -> x = y ) /\ Rel F ) ) $= ( wfunALTV ccoss cid wss wrel wa wbr wceq wal dffunALTV2 cossssid3 anbi1i cv wi bitri ) DEDFGHZDIZJCQZAQZDKUBBQZDKJUCUDLRBMAMCMZUAJDNTUEUAABCDOPS $. $} ${ $d F u x $. $( Alternate definition of the function relation predicate, cf. ~ dfdisjALTV4 . This is ~ dffun6 . For the ` X ` axis and the ` Y ` axis you can convert the right side to ` ( A. x1 E* y1 x1 F y1 /\ Rel F ) ` . (Contributed by NM, 9-Mar-1995.) $) dffunALTV4 $p |- ( FunALTV F <-> ( A. u E* x u F x /\ Rel F ) ) $= ( wfunALTV ccoss cid wss wrel wa cv wbr dffunALTV2 cossssid4 anbi1i bitri wmo wal ) CDCEFGZCHZIBJAJCKAPBQZSICLRTSABCMNO $. $} ${ $d F x y $. $( Alternate definition of the function relation predicate, cf. ~ dfdisjALTV5 . (Contributed by Peter Mazsa, 5-Sep-2021.) $) dffunALTV5 $p |- ( FunALTV F <-> ( A. x e. ran F A. y e. ran F ( x = y \/ ( [ x ] `' F i^i [ y ] `' F ) = (/) ) /\ Rel F ) ) $= ( wfunALTV ccoss cid wss wrel wa cv wceq ccnv cec cin crn wral dffunALTV2 c0 wo cossssid5 anbi1i bitri ) CDCEFGZCHZIAJZBJZKUECLZMUFUGMNRKSBCOZPAUHP ZUDICQUCUIUDABCTUAUB $. $} ${ $d F x $. $( Elementhood in the class of all functions. (Contributed by Peter Mazsa, 24-Jul-2021.) $) elfunsALTV $p |- ( F e. FunsALTV <-> ( ,~ F e. CnvRefRels /\ F e. Rels ) ) $= ( vx ccoss ccnvrefrels wcel crels cfunsALTV dffunsALTV wceq cosseq eleq1d cv rabeqel ) BLZCZDEACZDEBFGABHNAIOPDNAJKM $. $} $( Elementhood in the class of all functions. (Contributed by Peter Mazsa, 31-Aug-2021.) $) elfunsALTV2 $p |- ( F e. FunsALTV <-> ( ,~ F C_ _I /\ F e. Rels ) ) $= ( cfunsALTV wcel ccoss ccnvrefrels crels cid elfunsALTV cosselrels biantrud wa wss cosselcnvrefrels2 syl6rbbr pm5.32ri bitri ) ABCADZECZAFCZKQGLZSKAHSR TSTTQFCZKRSUATAFIJAMNOP $. ${ $d F u x y $. $( Elementhood in the class of all functions. (Contributed by Peter Mazsa, 31-Aug-2021.) $) elfunsALTV3 $p |- ( F e. FunsALTV <-> ( A. u A. x A. y ( ( u F x /\ u F y ) -> x = y ) /\ F e. Rels ) ) $= ( cfunsALTV wcel ccoss ccnvrefrels crels wa cv wbr wceq wi wal elfunsALTV cosselrels biantrud cosselcnvrefrels3 syl6rbbr pm5.32ri bitri ) DEFDGZHFZ DIFZJCKZAKZDLUFBKZDLJUGUHMNBOAOCOZUEJDPUEUDUIUEUIUIUCIFZJUDUEUJUIDIQRABCD STUAUB $. $} ${ $d F u x $. $( Elementhood in the class of all functions. (Contributed by Peter Mazsa, 31-Aug-2021.) $) elfunsALTV4 $p |- ( F e. FunsALTV <-> ( A. u E* x u F x /\ F e. Rels ) ) $= ( cfunsALTV wcel ccoss ccnvrefrels crels wa wbr wmo elfunsALTV cosselrels cv wal biantrud cosselcnvrefrels4 syl6rbbr pm5.32ri bitri ) CDECFZGEZCHEZ IBNANCJAKBOZUCICLUCUBUDUCUDUDUAHEZIUBUCUEUDCHMPABCQRST $. $} ${ $d F x y $. $( Elementhood in the class of all functions. (Contributed by Peter Mazsa, 5-Sep-2021.) $) elfunsALTV5 $p |- ( F e. FunsALTV <-> ( A. x e. ran F A. y e. ran F ( x = y \/ ( [ x ] `' F i^i [ y ] `' F ) = (/) ) /\ F e. Rels ) ) $= ( cfunsALTV wcel ccoss ccnvrefrels crels wa cv wceq ccnv cec cin crn wral c0 wo elfunsALTV cosselrels biantrud cosselcnvrefrels5 syl6rbbr pm5.32ri bitri ) CDECFZGEZCHEZIAJZBJZKUICLZMUJUKMNQKRBCOZPAULPZUHICSUHUGUMUHUMUMUF HEZIUGUHUNUMCHTUAABCUBUCUDUE $. $} $( The element of the class of all functions and the function predicate are the same when ` F ` is a set. (Contributed by Peter Mazsa, 26-Jul-2021.) $) elfunsALTVfunALTV $p |- ( F e. V -> ( F e. FunsALTV <-> FunALTV F ) ) $= ( wcel ccoss ccnvrefrels crels wa wcnvrefrel wrel cfunsALTV wfunALTV cvv wb cossex elcnvrefrelsrel syl elrelsrel anbi12d elfunsALTV df-funALTV 3bitr4g ) ABCZADZECZAFCZGUCHZAIZGAJCAKUBUDUFUEUGUBUCLCUDUFMABNUCLOPABQRASATUA $. $( Our definition of the function predicate ~ df-funALTV (based on a more general, converse reflexive, relation) and the original definition of function in set.mm ~ df-fun , are always the same and interchangeable. (Contributed by Peter Mazsa, 27-Jul-2021.) $) funALTVfun $p |- ( FunALTV F <-> Fun F ) $= ( wcnvrefrel wrel wa ccnv ccom cid wss wfunALTV wfun cnvrefrelcoss2 dfcoss3 ccoss sseq1i bitri anbi2ci df-funALTV df-fun 3bitr4i ) AMZBZACZDUBAAEFZGHZD AIAJUAUDUBUATGHUDAKTUCGALNOPAQARS $. $( Subclass theorem for function. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Mario Carneiro, 24-Jun-2014.) (Revised by Peter Mazsa, 22-Sep-2021.) $) funALTVss $p |- ( A C_ B -> ( FunALTV B -> FunALTV A ) ) $= ( wss ccoss cid wrel wa wfunALTV wi cossss sstr2 anim12d dffunALTV2 3imtr4g syl relss ) ABCZBDZECZBFZGADZECZAFZGBHAHQSUBTUCQUARCSUBIABJUAREKOABPLBMAMN $. $( Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.) $) funALTVeq $p |- ( A = B -> ( FunALTV A <-> FunALTV B ) ) $= ( wceq wfunALTV wss wi eqimss2 funALTVss syl eqimss impbid ) ABCZADZBDZLBAE MNFBAGBAHILABENMFABJABHIK $. ${ funALTVeqi.1 $e |- A = B $. $( Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) $) funALTVeqi $p |- ( FunALTV A <-> FunALTV B ) $= ( wceq wfunALTV wb funALTVeq ax-mp ) ABDAEBEFCABGH $. $} ${ funALTVeqd.1 $e |- ( ph -> A = B ) $. $( Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.) $) funALTVeqd $p |- ( ph -> ( FunALTV A <-> FunALTV B ) ) $= ( wceq wfunALTV wb funALTVeq syl ) ABCEBFCFGDBCHI $. $} $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Disjoints vs. converse functions =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) $( Define the class of all disjoint sets (but not necessarily disjoint relations, cf. ~ df-disjs ). It is used only by ~ df-disjs . (Contributed by Peter Mazsa, 17-Jul-2021.) $) df-disjss $a |- Disjss = { x | ,~ `' x e. CnvRefRels } $. $( Define the disjoint relations class, i.e., the class of all disjoints. We need ` Disjs ` for the definition of ` Parts ` and ` Part ` for the Partition-Equivalence Theorems: this need for ` Parts ` as disjoint relations on their domain quotients is the reason why we must define ` Disjs ` instead of simply using converse functions (cf. ~ dfdisjALTV ). The element of the class of all disjoints and the disjoint predicate are the same, that is ` ( R e. Disjs <-> Disj R ) ` when ` R ` is a set, cf. ~ eldisjsdisj . Alternate definitions are ~ dfdisjs , ... , ~ dfdisjs5 . (Contributed by Peter Mazsa, 17-Jul-2021.) $) df-disjs $a |- Disjs = ( Disjss i^i Rels ) $. $( Define the disjoint relation predicate, i.e., the disjoint predicate. A disjoint relation is a converse function of the relation by ~ dfdisjALTV , cf. the comment of ~ df-disjs why we need disjoint relations instead of converse functions anyway. The element of the class of all disjoints and the disjoint predicate are the same, that is ` ( R e. Disjs <-> Disj R ) ` when ` R ` is a set, cf. ~ eldisjsdisj . Alternate definitions are ~ dfdisjALTV , ... , ~ dfdisjALTV5 . (Contributed by Peter Mazsa, 17-Jul-2021.) $) df-disjALTV $a |- ( Disj R <-> ( CnvRefRel ,~ `' R /\ Rel R ) ) $. $( Define the disjoint elementhood relation predicate, i.e., the disjoint elementhood predicate. Read: the elements of ` A ` are disjoint. Cf. the comment of ~ df-eleqvrel . As of now, disjoint elementhood is defined as "partition" in set.mm : compare ~ df-prt with ~ dfeldisj5 . Cf. the comments of ~ dfmembpart2 and of ~ df-parts . (Contributed by Peter Mazsa, 17-Jul-2021.) $) df-eldisj $a |- ( ElDisj A <-> Disj ( `' _E |` A ) ) $. $( Alternate definition of the class of all disjoints. (Contributed by Peter Mazsa, 18-Jul-2021.) $) dfdisjs $p |- Disjs = { r e. Rels | ,~ `' r e. CnvRefRels } $= ( cv ccnv ccoss ccnvrefrels cdisjs cdisjss crels df-disjs df-disjss abeqin wcel ) ABCDELAFGHIAJK $. $( Alternate definition of the class of all disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.) $) dfdisjs2 $p |- Disjs = { r e. Rels | ,~ `' r C_ _I } $= ( cv ccnv ccoss ccnvrefrels wcel cid wss crels cdisjs dfdisjs cosscnvelrels wa biantrud cosselcnvrefrels2 syl6rbbr rabimbieq ) ABZCZDZEFZTGHZAIJAKRIFZU BUBTIFZMUAUCUDUBRILNSOPQ $. ${ $d r u v x $. $( Alternate definition of the class of all disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.) $) dfdisjs3 $p |- Disjs = { r e. Rels | A. u A. v A. x ( ( u r x /\ v r x ) -> u = v ) } $= ( cdisjs cv ccnv ccoss cid wss crels crab wbr wa wceq wi wal cosscnvssid3 dfdisjs2 rabbiia2 eqtri ) EDFZGHIJZDKLCFZAFZUBMBFZUEUBMNUDUFOPAQBQCQZDKLD SUCUGDKABCUBRTUA $. $} ${ $d r u x $. $( Alternate definition of the class of all disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.) $) dfdisjs4 $p |- Disjs = { r e. Rels | A. x E* u u r x } $= ( cdisjs ccnv ccoss cid wss crels crab wbr dfdisjs2 cosscnvssid4 rabbiia2 cv wmo wal eqtri ) DCOZEFGHZCIJBOAOSKBPAQZCIJCLTUACIABSMNR $. $} ${ $d r u v $. $( Alternate definition of the class of all disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.) $) dfdisjs5 $p |- Disjs = { r e. Rels | A. u e. dom r A. v e. dom r ( u = v \/ ( [ u ] r i^i [ v ] r ) = (/) ) } $= ( cv ccnv ccoss cid wss wceq cec cin c0 wo cdm wral crels cdisjs biantrud wb wa dfdisjs2 wcel cosscnvssid5 elrelsrelim bibi12d mpbiri rabimbieq wrel ) CDZEFGHZBDZADZIUKUIJULUIJKLIMAUINZOBUMOZCPQCUAUIPUBZUJUNSUJUIUHZTZ UNUPTZSABUIUCUOUJUQUNURUOUPUJUIUDZRUOUPUNUSRUEUFUG $. $} $( Alternate definition of the disjoint relation predicate. A disjoint relation is a converse function of the relation, cf. the comment of ~ df-disjs why we need disjoint relations instead of converse functions anyway. (Contributed by Peter Mazsa, 27-Jul-2021.) $) dfdisjALTV $p |- ( Disj R <-> ( FunALTV `' R /\ Rel R ) ) $= ( wdisjALTV ccnv ccoss wcnvrefrel wa wfunALTV df-disjALTV relcnv df-funALTV wrel mpbiran2 anbi1i bitr4i ) ABACZDEZAKZFOGZQFAHRPQRPOKAIOJLMN $. $( Alternate definition of the disjoint relation predicate, cf. ~ dffunALTV2 . (Contributed by Peter Mazsa, 27-Jul-2021.) $) dfdisjALTV2 $p |- ( Disj R <-> ( ,~ `' R C_ _I /\ Rel R ) ) $= ( wdisjALTV ccnv ccoss wcnvrefrel wrel wa df-disjALTV cnvrefrelcoss2 anbi1i cid wss bitri ) ABACZDZEZAFZGOKLZQGAHPRQNIJM $. ${ $d R u v x $. $( Alternate definition of the disjoint relation predicate, cf. ~ dffunALTV3 . (Contributed by Peter Mazsa, 28-Jul-2021.) $) dfdisjALTV3 $p |- ( Disj R <-> ( A. u A. v A. x ( ( u R x /\ v R x ) -> u = v ) /\ Rel R ) ) $= ( wdisjALTV ccnv ccoss cid wss wrel wa cv wbr wi dfdisjALTV2 cosscnvssid3 wceq wal anbi1i bitri ) DEDFGHIZDJZKCLZALZDMBLZUDDMKUCUEQNARBRCRZUBKDOUAU FUBABCDPST $. $} ${ $d R u x $. $( Alternate definition of the disjoint relation predicate, cf. ~ dffunALTV4 . (Contributed by Peter Mazsa, 5-Sep-2021.) $) dfdisjALTV4 $p |- ( Disj R <-> ( A. x E* u u R x /\ Rel R ) ) $= ( wdisjALTV ccnv ccoss cid wss wa cv wbr wmo wal dfdisjALTV2 cosscnvssid4 wrel anbi1i bitri ) CDCEFGHZCPZIBJAJCKBLAMZTICNSUATABCOQR $. $} ${ $d R u v $. $( Alternate definition of the disjoint relation predicate, cf. ~ dffunALTV5 . (Contributed by Peter Mazsa, 5-Sep-2021.) $) dfdisjALTV5 $p |- ( Disj R <-> ( A. u e. dom R A. v e. dom R ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) /\ Rel R ) ) $= ( wdisjALTV ccnv ccoss cid wss wrel wa cv wceq cec cin c0 cdm dfdisjALTV2 wo wral cosscnvssid5 bitri ) CDCEFGHCIZJBKZAKZLUCCMUDCMNOLRACPZSBUESUBJCQ ABCTUA $. $} $( Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.) $) dfeldisj2 $p |- ( ElDisj A <-> ,~ `' ( `' _E |` A ) C_ _I ) $= ( weldisj cep ccnv cres wdisjALTV cid wss df-eldisj wrel relres dfdisjALTV2 ccoss mpbiran2 bitri ) ABCDZAEZFZQDMGHZAIRSQJPAKQLNO $. ${ $d A u v x $. $( Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.) $) dfeldisj3 $p |- ( ElDisj A <-> A. u e. A A. v e. A A. x e. ( u i^i v ) u = v ) $= ( weldisj cv wcel cin w3a wceq wi wal wral wbr cvv brcnvepres el2v bitr4i wa wb cep ccnv cres wdisjALTV df-eldisj wrel dfdisjALTV3 mpbiran2 anbi12i relres an4 elin anbi2i 3bitr4i df-3an imbi1i 3albii 3bitri r3al ) DEZCFZD GZBFZDGZAFZVAVCHZGZIZVAVCJZKZALBLCLZVIAVFMBDMCDMUTUAUBZDUCZUDZVAVEVMNZVCV EVMNZSZVIKZALBLCLZVKDUEVNVSVMUFVLDUJABCVMUGUHVRVJCBAVQVHVIVQVBVDSZVGSZVHV BVEVAGZSZVDVEVCGZSZSVTWBWDSZSVQWAVBWBVDWDUKVOWCVPWEVOWCTCADVAVEOOPQVPWETB ADVCVEOOPQUIVGWFVTVEVAVCULUMUNVBVDVGUORUPUQURVICBADDVFUSR $. $} ${ $d A u x $. $( Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.) $) dfeldisj4 $p |- ( ElDisj A <-> A. x E* u e. A x e. u ) $= ( weldisj cep ccnv cres wdisjALTV cv wbr wmo wal wcel wrmo df-eldisj wrel relres dfdisjALTV4 mpbiran2 cvv wa wb brcnvepres el2v mobii df-rmo bitr4i albii 3bitri ) CDEFZCGZHZBIZAIZUKJZBKZALZUNUMMZBCNZALCOULUQUKPUJCQABUKRSU PUSAUPUMCMURUAZBKUSUOUTBUOUTUBBACUMUNTTUCUDUEURBCUFUGUHUI $. $} ${ $d A u v x $. $( Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.) $) dfeldisj5 $p |- ( ElDisj A <-> A. u e. A A. v e. A ( u = v \/ ( u i^i v ) = (/) ) ) $= ( vx weldisj cv wrmo wal wceq cin c0 wral cep cec biantru cvv eccnvep elv wo wa wcel dfeldisj4 ccnv wbr wrel inecmo2 relcnv 3bitr4i ineq12i 2ralbii eqeq1i orbi2i wb brcnvep rmobii albii 3bitr3i bitr4i ) CEDFZBFZUAZBCGZDHZ UTAFZIZUTVDJZKIZSZACLBCLZDBCUBVEUTMUCZNZVDVJNZJZKIZSZACLBCLZUTUSVJUDZBCGZ DHZVIVCVPVJUEZTVSVTTVPVSDABCVJUFVTVPMUGZOVTVSWAOUHVOVHBACCVNVGVEVMVFKVKUT VLVDVKUTIBUTPQRVLVDIAVDPQRUIUKULUJVRVBDVQVABCVQVAUMBUTUSPUNRUOUPUQUR $. $} ${ $d R r $. $( Elementhood in the class of all disjoints. (Contributed by Peter Mazsa, 24-Jul-2021.) $) eldisjs $p |- ( R e. Disjs <-> ( ,~ `' R e. CnvRefRels /\ R e. Rels ) ) $= ( vr ccnv ccoss ccnvrefrels wcel crels cdisjs dfdisjs wceq cosseqd eleq1d cv cnveq rabeqel ) BMZCZDZEFACZDZEFBGHABIPAJZRTEUAQSPANKLO $. $} $( Elementhood in the class of all disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.) $) eldisjs2 $p |- ( R e. Disjs <-> ( ,~ `' R C_ _I /\ R e. Rels ) ) $= ( cdisjs wcel ccnv ccoss ccnvrefrels crels wa cid wss eldisjs cosscnvelrels biantrud cosselcnvrefrels2 syl6rbbr pm5.32ri bitri ) ABCADZEZFCZAGCZHSIJZUA HAKUATUBUAUBUBSGCZHTUAUCUBAGLMRNOPQ $. ${ $d R u v x $. $( Elementhood in the class of all disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.) $) eldisjs3 $p |- ( R e. Disjs <-> ( A. u A. v A. x ( ( u R x /\ v R x ) -> u = v ) /\ R e. Rels ) ) $= ( cdisjs wcel ccnv ccoss cid wss crels wa cv wbr wceq wi wal cosscnvssid3 eldisjs2 anbi1i bitri ) DEFDGHIJZDKFZLCMZAMZDNBMZUEDNLUDUFOPAQBQCQZUCLDSU BUGUCABCDRTUA $. $} ${ $d R u x $. $( Elementhood in the class of all disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.) $) eldisjs4 $p |- ( R e. Disjs <-> ( A. x E* u u R x /\ R e. Rels ) ) $= ( cdisjs wcel ccoss cid wss crels wa cv wbr wmo wal eldisjs2 cosscnvssid4 ccnv anbi1i bitri ) CDECQFGHZCIEZJBKAKCLBMANZUAJCOTUBUAABCPRS $. $} ${ $d R u v $. $( Elementhood in the class of all disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.) $) eldisjs5 $p |- ( R e. V -> ( R e. Disjs <-> ( A. u e. dom R A. v e. dom R ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) /\ R e. Rels ) ) ) $= ( cdisjs wcel ccnv ccoss cid wss crels wa cv wceq cec cin c0 wral anbi2d wb wo cdm eldisjs2 wrel cosscnvssid5 elrelsrel bibi12d mpbiri syl5bb ) CE FCGHIJZCKFZLZCDFZBMZAMZNUNCOUOCOPQNUAACUBZRBUPRZUKLZCUCUMULURTUJCUDZLZUQU SLZTABCUEUMULUTURVAUMUKUSUJCDUFZSUMUKUSUQVBSUGUHUI $. $} $( The element of the class of all disjoint relations and the disjoint relation predicate are the same, that is ` ( R e. Disjs <-> Disj R ) ` when ` R ` is a set. (Contributed by Peter Mazsa, 25-Jul-2021.) $) eldisjsdisj $p |- ( R e. V -> ( R e. Disjs <-> Disj R ) ) $= ( wcel ccnv ccoss ccnvrefrels crels wa wcnvrefrel wrel cdisjs wdisjALTV cvv wb cosscnvex elcnvrefrelsrel syl elrelsrel anbi12d eldisjs df-disjALTV 3bitr4g ) ABCZADEZFCZAGCZHUDIZAJZHAKCALUCUEUGUFUHUCUDMCUEUGNABOUDMPQABRSATA UAUB $. $( Disjoint relation is a relation. (Contributed by Peter Mazsa, 15-Sep-2021.) $) disjrel $p |- ( Disj R -> Rel R ) $= ( wdisjALTV ccnv ccoss wcnvrefrel wrel df-disjALTV simprbi ) ABACDEAFAGH $. $( Subclass theorem for disjoints. (Contributed by Peter Mazsa, 28-Oct-2020.) (Revised by Peter Mazsa, 22-Sep-2021.) $) disjss $p |- ( A C_ B -> ( Disj B -> Disj A ) ) $= ( wss ccnv wfunALTV wrel wa wdisjALTV wi cnvss funALTVss anim12d dfdisjALTV syl relss 3imtr4g ) ABCZBDZEZBFZGADZEZAFZGBHAHQSUBTUCQUARCSUBIABJUARKNABOLB MAMP $. ${ disjssi.1 $e |- A C_ B $. $( Subclass theorem for disjoints, inference version. (Contributed by Peter Mazsa, 28-Sep-2021.) $) disjssi $p |- ( Disj B -> Disj A ) $= ( wss wdisjALTV wi disjss ax-mp ) ABDBEAEFCABGH $. $} ${ disjssd.1 $e |- ( ph -> A C_ B ) $. $( Subclass theorem for disjoints, deduction version. (Contributed by Peter Mazsa, 28-Sep-2021.) $) disjssd $p |- ( ph -> ( Disj B -> Disj A ) ) $= ( wss wdisjALTV wi disjss syl ) ABCECFBFGDBCHI $. $} $( Equality theorem for disjoints. (Contributed by Peter Mazsa, 22-Sep-2021.) $) disjeq $p |- ( A = B -> ( Disj A <-> Disj B ) ) $= ( wceq wdisjALTV wss wi eqimss2 disjss syl eqimss impbid ) ABCZADZBDZLBAEMN FBAGBAHILABENMFABJABHIK $. ${ disjeqi.1 $e |- A = B $. $( Equality theorem for disjoints, inference version. (Contributed by Peter Mazsa, 22-Sep-2021.) $) disjeqi $p |- ( Disj A <-> Disj B ) $= ( wceq wdisjALTV wb disjeq ax-mp ) ABDAEBEFCABGH $. $} ${ disjeqd.1 $e |- ( ph -> A = B ) $. $( Equality theorem for disjoints, deduction version. (Contributed by Peter Mazsa, 22-Sep-2021.) $) disjeqd $p |- ( ph -> ( Disj A <-> Disj B ) ) $= ( wceq wdisjALTV wb disjeq syl ) ABCEBFCFGDBCHI $. $} $( Lemma for the equality theorem for partition ~ parteq1 . (Contributed by Peter Mazsa, 5-Oct-2021.) $) disjdmqseqeq1 $p |- ( R = S -> ( ( Disj R /\ ( dom R /. R ) = A ) <-> ( Disj S /\ ( dom S /. S ) = A ) ) ) $= ( wceq wdisjALTV cdm cqs disjeq dmqseqeq1 anbi12d ) BCDBECEBFBGADCFCGADBCHA BCIJ $. $( Subclass theorem for disjoint elementhood. (Contributed by Peter Mazsa, 23-Sep-2021.) $) eldisjss $p |- ( A C_ B -> ( ElDisj B -> ElDisj A ) ) $= ( wss cep ccnv cres wdisjALTV weldisj ssres2 disjss syl df-eldisj 3imtr4g wi ) ABCZDEZBFZGZPAFZGZBHAHOSQCRTNABPISQJKBLALM $. ${ eldisjssi.1 $e |- A C_ B $. $( Subclass theorem for disjoint elementhood, inference version. (Contributed by Peter Mazsa, 28-Sep-2021.) $) eldisjssi $p |- ( ElDisj B -> ElDisj A ) $= ( wss weldisj wi eldisjss ax-mp ) ABDBEAEFCABGH $. $} ${ eldisjssd.1 $e |- ( ph -> A C_ B ) $. $( Subclass theorem for disjoint elementhood, deduction version. (Contributed by Peter Mazsa, 28-Sep-2021.) $) eldisjssd $p |- ( ph -> ( ElDisj B -> ElDisj A ) ) $= ( wss weldisj wi eldisjss syl ) ABCECFBFGDBCHI $. $} $( Equality theorem for disjoint elementhood. (Contributed by Peter Mazsa, 23-Sep-2021.) $) eldisjeq $p |- ( A = B -> ( ElDisj A <-> ElDisj B ) ) $= ( wceq cep ccnv cres wdisjALTV weldisj reseq2 disjeq syl df-eldisj 3bitr4g wb ) ABCZDEZAFZGZPBFZGZAHBHOQSCRTNABPIQSJKALBLM $. ${ eldisjeqi.1 $e |- A = B $. $( Equality theorem for disjoint elementhood, inference version. (Contributed by Peter Mazsa, 23-Sep-2021.) $) eldisjeqi $p |- ( ElDisj A <-> ElDisj B ) $= ( wceq weldisj wb eldisjeq ax-mp ) ABDAEBEFCABGH $. $} ${ eldisjeqd.1 $e |- ( ph -> A = B ) $. $( Equality theorem for disjoint elementhood, deduction version. (Contributed by Peter Mazsa, 23-Sep-2021.) $) eldisjeqd $p |- ( ph -> ( ElDisj A <-> ElDisj B ) ) $= ( wceq weldisj wb eldisjeq syl ) ABCEBFCFGDBCHI $. $} $( Two ways of saying that a tail Cartesian product is disjoint. (Contributed by Peter Mazsa, 17-Jun-2020.) (Revised by Peter Mazsa, 21-Sep-2021.) $) disjtxp $p |- ( Disj ( R (x) S ) <-> ( ,~ `' R i^i ,~ `' S ) C_ _I ) $= ( ctxp wdisjALTV ccnv ccoss cid wss cin wrel txprel dfdisjALTV2 1cosscnvtxp mpbiran2 sseq1i bitri ) ABCZDZQEFZGHZAEFBEFIZGHRTQJABKQLNSUAGABMOP $. $( Disjointness condition for tail Cartesian product. (Contributed by Peter Mazsa, 12-Jul-2020.) (Revised by Peter Mazsa, 22-Sep-2021.) $) disjorimtxp $p |- ( ( Disj R \/ Disj S ) -> Disj ( R (x) S ) ) $= ( wdisjALTV wo ccnv ccoss cin cid wss ctxp wrel dfdisjALTV2 simplbi orim12i inss syl disjtxp sylibr ) ACZBCZDZAEFZBEFZGHIZABJCUAUBHIZUCHIZDUDSUETUFSUEA KALMTUFBKBLMNUBUCHOPABQR $. $( Disjointness condition for tail Cartesian product. (Contributed by Peter Mazsa, 15-Dec-2020.) (Revised by Peter Mazsa, 22-Sep-2021.) $) disjimtxp $p |- ( Disj S -> Disj ( R (x) S ) ) $= ( wdisjALTV ctxp disjorimtxp olcs ) ACBCABDCABEF $. $( Disjointness condition for restriction. (Contributed by Peter Mazsa, 27-Sep-2021.) $) disjimres $p |- ( Disj R -> Disj ( R |` A ) ) $= ( cres resss disjssi ) BACBBADE $. $( Disjointness condition for intersection. (Contributed by Peter Mazsa, 11-Jun-2021.) (Revised by Peter Mazsa, 28-Sep-2021.) $) disjimin $p |- ( Disj S -> Disj ( R i^i S ) ) $= ( cin inss2 disjssi ) ABCBABDE $. $( Disjointness condition for tail Cartesian product with restriction. (Contributed by Peter Mazsa, 27-Sep-2021.) $) disjimtxpres $p |- ( Disj S -> Disj ( R (x) ( S |` A ) ) ) $= ( wdisjALTV cres ctxp disjimres disjimtxp syl ) CDCAEZDBJFDACGBJHI $. $( Disjointness condition for tail Cartesian product with restriction. (Contributed by Peter Mazsa, 27-Sep-2021.) $) disjiminres $p |- ( Disj S -> Disj ( R i^i ( S |` A ) ) ) $= ( wdisjALTV cres cin disjimres disjimin syl ) CDCAEZDBJFDACGBJHI $. ${ $d u x $. $( The null class is disjoint. (Contributed by Peter Mazsa, 27-Sep-2021.) $) disjALTV0 $p |- Disj (/) $= ( vu vx c0 wdisjALTV cv wbr wmo wal wrel wex wn br0 nex nexmo ax-gen rel0 ax-mp dfdisjALTV4 mpbir2an ) CDAEZBEZCFZAGZBHCIUCBUBAJKUCUBATUALMUBANQOPB ACRS $. $} $( The class of all identity relations is disjoint. (Contributed by Peter Mazsa, 20-Jun-2021.) $) disjALTVid $p |- Disj _I $= ( cid wdisjALTV ccnv ccoss wrel cosscnvid eqimssi reli dfdisjALTV2 mpbir2an wss ) ABACDZAKAELAFGHAIJ $. $( The class of all identity relations restricted is disjoint. (Contributed by Peter Mazsa, 28-Jun-2020.) (Revised by Peter Mazsa, 27-Sep-2021.) $) disjALTVidres $p |- Disj ( _I |` A ) $= ( cid wdisjALTV cres disjALTVid disjimres ax-mp ) BCBADCEABFG $. $( The intersection with restricted identity relation is disjoint. (Contributed by Peter Mazsa, 31-Dec-2021.) $) disjALTVinidres $p |- Disj ( R i^i ( _I |` A ) ) $= ( cid wdisjALTV cres cin disjALTVid disjiminres ax-mp ) CDBCAEFDGABCHI $. $( The class of tail Cartesian product with restricted identity relation is disjoint. (Contributed by Peter Mazsa, 25-Jun-2020.) (Revised by Peter Mazsa, 27-Sep-2021.) $) disjALTVtxpidres $p |- Disj ( R (x) ( _I |` A ) ) $= ( cid wdisjALTV cres ctxp disjALTVid disjimtxpres ax-mp ) CDBCAEFDGABCHI $. $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Partitions: disjoints on domain quotients =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) $( Define the class of all partitions, cf. the comment of ~ df-disjs . Partitions are disjoints on domain quotients (or: domain quotients restricted to disjoints). This is a more general meaning of partition than we accustomed to: the conventional meaning of partition (e.g. partition ` A ` of ` X ` , [Halmos] p. 28: "A partition of ` X ` is a disjoint collection ` A ` of non-empty subsets of ` X ` whose union is ` X ` ", or Definition 35, [Suppes] p. 83., cf. https://oeis.org/A000110 ) is what we call here membership partition, cf. ~ dfmembpart2 . The binary partitions relation and the partition predicate are the same, that is ` ( R Parts A <-> R Part A ) ` when ` A ` and ` R ` are sets, cf. ~ brpartspart . (Contributed by Peter Mazsa, 26-Jun-2021.) $) df-parts $a |- Parts = ( DomainQss |` Disjs ) $. $( Define the partition predicate (read: ` A ` is a partition by ` R ` ). Alternate definition is ~ dfpart2 . Binary partition and the partition predicate are the same when ` A ` and ` R ` are sets, cf. ~ brpartspart . (Contributed by Peter Mazsa, 12-Aug-2021.) $) df-part $a |- ( R Part A <-> ( Disj R /\ R DomainQs A ) ) $. $( Define the membership partition predicate, or the disjoint restricted elementhood relation on its domain quotient predicate. (Read: ` A ` is a membership partition.) Alternate definition is ~ dfmembpart2 . Membership partition is the conventional meaning of partition (cf. the notes of ~ df-parts and ~ dfmembpart2 ), we generalize the concept in ~ df-parts and ~ df-part . Membership partition and membership equivalence are the same by ~ mpet . (Contributed by Peter Mazsa, 26-Jun-2021.) $) df-membpart $a |- ( MembPart A <-> ( `' _E |` A ) Part A ) $. $( Alternate definition of the partition predicate. (Contributed by Peter Mazsa, 5-Sep-2021.) $) dfpart2 $p |- ( R Part A <-> ( Disj R /\ ( dom R /. R ) = A ) ) $= ( wpart wdisjALTV wdmqs wa cdm cqs wceq df-part df-dmqs anbi2i bitri ) ABCB DZABEZFNBGBHAIZFABJOPNABKLM $. $( Alternate definition of the conventional membership case of partition. Partition ` A ` of ` X ` , [Halmos] p. 28: "A partition of ` X ` is a disjoint collection ` A ` of non-empty subsets of ` X ` whose union is ` X ` ", or Definition 35, [Suppes] p. 83., cf. https://oeis.org/A000110 . (Contributed by Peter Mazsa, 14-Aug-2021.) $) dfmembpart2 $p |- ( MembPart A <-> ( ElDisj A /\ -. (/) e. A ) ) $= ( wmembpart cep ccnv cres wpart wdisjALTV wdmqs wa weldisj wcel df-membpart c0 wn df-part df-eldisj bicomi cnvepresdmqs anbi12i 3bitri ) ABACDAEZFUAGZA UAHZIAJZMAKNZIALAUAOUBUDUCUEUDUBAPQARST $. $( Binary partitions relation. (Contributed by Peter Mazsa, 23-Jul-2021.) $) brparts $p |- ( A e. V -> ( R Parts A <-> ( R e. Disjs /\ R DomainQss A ) ) ) $= ( cdisjs cparts cdmqss df-parts eqres ) BADEFCGH $. $( Binary partitions relation. (Contributed by Peter Mazsa, 30-Dec-2021.) $) brparts2 $p |- ( ( A e. V /\ R e. W ) -> ( R Parts A <-> ( R e. Disjs /\ ( dom R /. R ) = A ) ) ) $= ( wcel wa cparts wbr cdisjs cdmqss cdm cqs wb brparts adantr brdmqss anbi2d wceq bitrd ) ACEZBDEZFZBAGHZBIEZBAJHZFZUDBKBLARZFTUCUFMUAABCNOUBUEUGUDABCDP QS $. $( Binary partition and the partition predicate are the same when ` A ` and ` R ` are sets. (Contributed by Peter Mazsa, 5-Sep-2021.) $) brpartspart $p |- ( ( A e. V /\ R e. W ) -> ( R Parts A <-> R Part A ) ) $= ( wcel wa cparts wdisjALTV wdmqs wpart cdisjs cdmqss wb brparts eldisjsdisj wbr adantr adantl brdmqssqs anbi12d bitrd df-part a1i bitr4d ) ACEZBDEZFZBA GPZBHZABIZFZABJZUGUHBKEZBALPZFZUKUEUHUOMUFABCNQUGUMUIUNUJUFUMUIMUEBDORABCDS TUAULUKMUGABUBUCUD $. $( Equality theorem for partition. (Contributed by Peter Mazsa, 5-Oct-2021.) $) parteq1 $p |- ( R = S -> ( R Part A <-> S Part A ) ) $= ( wceq wdisjALTV cdm cqs wa wpart disjdmqseqeq1 dfpart2 3bitr4g ) BCDBEBFBG ADHCECFCGADHABIACIABCJABKACKL $. ${ parteq1i.1 $e |- R = S $. $( Equality theorem for partition, inference version. (Contributed by Peter Mazsa, 5-Oct-2021.) $) parteq1i $p |- ( R Part A <-> S Part A ) $= ( wceq wpart wb parteq1 ax-mp ) BCEABFACFGDABCHI $. $} ${ parteq1d.1 $e |- ( ph -> R = S ) $. $( Equality theorem for partition, deduction version. (Contributed by Peter Mazsa, 5-Oct-2021.) $) parteq1d $p |- ( ph -> ( R Part A <-> S Part A ) ) $= ( wceq wpart wb parteq1 syl ) ACDFBCGBDGHEBCDIJ $. $} $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Partition-Equivalence Theorems =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) ${ $d R u x y z $. $( The Teorem of "Divide et Aequivalere": every disjoint relation generates equivalent cosets by the relation: generalization of the former ~ prter1 , cf. ~ mdisjim . (Contributed by Peter Mazsa, 3-May-2019.) (Revised by Peter Mazsa, 17-Sep-2021.) $) disjim $p |- ( Disj R -> EqvRel ,~ R ) $= ( vx vy vz vu wdisjALTV cv ccoss wbr wal weqvrel wrel dfdisjALTV4 simplbi wa wi wmo trcoss syl eqvrelcoss3 sylibr ) AFZBGZCGZAHZIUDDGZUEIOUCUFUEIPD JCJBJZUEKUBEGUDAIEQCJZUGUBUHALCEAMNBCDEARSBCDATUA $. $} ${ disjimi.1 $e |- Disj R $. $( Every disjoint relation generates equivalent cosets by the relation, inference version. (Contributed by Peter Mazsa, 30-Sep-2021.) $) disjimi $p |- EqvRel ,~ R $= ( wdisjALTV ccoss weqvrel disjim ax-mp ) ACADEBAFG $. $} ${ disjimd.1 $e |- ( ph -> Disj R ) $. $( Every disjoint relation generates equivalent cosets by the relation, deduction version. (Contributed by Peter Mazsa, 30-Sep-2021.) $) disjimd $p |- ( ph -> EqvRel ,~ R ) $= ( wdisjALTV ccoss weqvrel disjim syl ) ABDBEFCBGH $. $} ${ detlem.1 $e |- Disj R $. $( If a relation is disjoint, then it is equivalent to the equivalent cosets by the relation, inference version. (Contributed by Peter Mazsa, 30-Sep-2021.) $) detlem $p |- ( Disj R <-> EqvRel ,~ R ) $= ( wdisjALTV ccoss weqvrel disjim a1i impbii ) ACZADEZAFIJBGH $. $} ${ detlemd.1 $e |- ( ph -> Disj R ) $. $( If a relation is disjoint, then it is equivalent to the equivalent cosets by the relation, deduction version. (Contributed by Peter Mazsa, 30-Sep-2021.) $) detlemd $p |- ( ph -> ( Disj R <-> EqvRel ,~ R ) ) $= ( wdisjALTV ccoss weqvrel disjim a1d impbid2 ) ABDZBEFZBGAJKCHI $. $} $( If the elements of ` A ` are disjoint, then it has equivalent coelements (former ~ prter1 ). Special case of ~ disjim . (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015) (Revised by Peter Mazsa, 8-Feb-2018.) ( Revised by Peter Mazsa, 23-Sep-2021.) $) mdisjim $p |- ( ElDisj A -> EqvRel ~ A ) $= ( cep ccnv wdisjALTV ccoss weqvrel weldisj ccoels disjim df-eldisj df-coels cres eqvreleqi 3imtr4i ) BCALZDOEZFAGAHZFOIAJQPAKMN $. ${ mdisjimi.1 $e |- ElDisj A $. $( If the elements of ` A ` are disjoint, then it has equivalent coelements, inference version. (Contributed by Peter Mazsa, 30-Sep-2021.) $) mdisjimi $p |- EqvRel ~ A $= ( weldisj ccoels weqvrel mdisjim ax-mp ) ACADEBAFG $. $} ${ mdisjimd.1 $e |- ( ph -> ElDisj A ) $. $( If the elements of ` A ` are disjoint, then it has equivalent coelements, deduction version. (Contributed by Peter Mazsa, 30-Sep-2021.) $) mdisjimd $p |- ( ph -> EqvRel ~ A ) $= ( weldisj ccoels weqvrel mdisjim syl ) ABDBEFCBGH $. $} ${ mdetlemi.1 $e |- ElDisj A $. $( If the elements of ` A ` are disjoint, then it is equivalent to the equivalent coelements of it, inference version. (Contributed by Peter Mazsa, 30-Sep-2021.) $) mdetlemi $p |- ( ElDisj A <-> EqvRel ~ A ) $= ( weldisj ccoels weqvrel mdisjim a1i impbii ) ACZADEZAFIJBGH $. $} ${ mdetlemd.1 $e |- ( ph -> ElDisj A ) $. $( If the elements of ` A ` are disjoint, then it is equivalent to the equivalent coelements of it, deduction version. (Contributed by Peter Mazsa, 30-Sep-2021.) $) mdetlemd $p |- ( ph -> ( ElDisj A <-> EqvRel ~ A ) ) $= ( weldisj ccoels weqvrel mdisjim a1d impbid2 ) ABDZBEFZBGAJKCHI $. $} $( The null class is an equivalence relation. (Contributed by Peter Mazsa, 31-Dec-2021.) $) eqvrel0 $p |- EqvRel (/) $= ( c0 ccoss weqvrel disjALTV0 disjimi coss0 eqvreleqi mpbi ) ABZCACADEIAFGH $. $( The cosets by the null class is an equivalence relation if and only if the null class is disjoint. (Contributed by Peter Mazsa, 31-Dec-2021.) $) det0 $p |- ( Disj (/) <-> EqvRel ,~ (/) ) $= ( c0 disjALTV0 detlem ) ABC $. $( Identity relation is an equivalence relation. (Contributed by Peter Mazsa, 15-Apr-2019.) (Revised by Peter Mazsa, 31-Dec-2021.) $) eqvrelid $p |- EqvRel _I $= ( cid ccoss weqvrel disjALTVid disjimi cossid eqvreleqi mpbi ) ABZCACADEIAF GH $. $( The cosets by a restricted identity relation is an equivalence relation. (Contributed by Peter Mazsa, 31-Dec-2021.) $) eqvrel1cossidres $p |- EqvRel ,~ ( _I |` A ) $= ( cid cres disjALTVidres disjimi ) BACADE $. $( The cosets by an intersection with a restricted identity relation is an equivalence relation. (Contributed by Peter Mazsa, 31-Dec-2021.) $) eqvrel1cossinidres $p |- EqvRel ,~ ( R i^i ( _I |` A ) ) $= ( cid cres cin disjALTVinidres disjimi ) BCADEABFG $. $( The cosets by a tail Cartesian product with a restricted identity relation is an equivalence relation. (Contributed by Peter Mazsa, 31-Dec-2021.) $) eqvrel1cosstxpidres $p |- EqvRel ,~ ( R (x) ( _I |` A ) ) $= ( cid cres ctxp disjALTVtxpidres disjimi ) BCADEABFG $. $( The cosets by the identity relation is an equivalence relation if and only if the identity relation is disjoint. (Contributed by Peter Mazsa, 31-Dec-2021.) $) detid $p |- ( Disj _I <-> EqvRel ,~ _I ) $= ( cid disjALTVid detlem ) ABC $. $( The cosets by the restricted identity relation is an equivalence relation if and only if the restricted identity relation is disjoint. (Contributed by Peter Mazsa, 31-Dec-2021.) $) detidres $p |- ( Disj ( _I |` A ) <-> EqvRel ,~ ( _I |` A ) ) $= ( cid cres disjALTVidres detlem ) BACADE $. $( The cosets by the intersection with the restricted identity relation is an equivalence relation if and only if the intersection with the restricted identity relation is disjoint. (Contributed by Peter Mazsa, 31-Dec-2021.) $) detinidres $p |- ( Disj ( R i^i ( _I |` A ) ) <-> EqvRel ,~ ( R i^i ( _I |` A ) ) ) $= ( cid cres cin disjALTVinidres detlem ) BCADEABFG $. $( The cosets by the tail Cartesian product with the restricted identity relation is an equivalence relation if and only if the tail Cartesian product with the restricted identity relation is disjoint. (Contributed by Peter Mazsa, 31-Dec-2021.) $) dettxpidres $p |- ( Disj ( R (x) ( _I |` A ) ) <-> EqvRel ,~ ( R (x) ( _I |` A ) ) ) $= ( cid cres ctxp disjALTVtxpidres detlem ) BCADEABFG $. ${ $d R x y $. $( Lemma for ~ disjdmqseq , ~ partim2 and ~ petlem via ~ disjlem17 , (general version of the former ~ prtlem14 ). (Contributed by Peter Mazsa, 10-Sep-2021.) $) disjlem14 $p |- ( Disj R -> ( ( x e. dom R /\ y e. dom R ) -> ( ( A e. [ x ] R /\ A e. [ y ] R ) -> [ x ] R = [ y ] R ) ) ) $= ( wdisjALTV cv cdm wcel wa wceq cec cin c0 wo wi wral dfdisjALTV5 simplbi wrel rsp2 syl eceq1 a1d elin nel02 pm2.21d syl5bir jaoi syl6 ) DEZAFZDGZH BFZULHIZUKUMJZUKDKZUMDKZLZMJZNZCUPHCUQHIZUPUQJZOZUJUTBULPAULPZUNUTOUJVDDS BADQRUTABULULTUAUOVCUSUOVBVAUKUMDUBUCVACURHZUSVBCUPUQUDUSVEVBURCUEUFUGUHU I $. $} ${ $d A y $. $d B y $. $d R x y $. $( Lemma for ~ disjdmqseq , ~ partim2 and ~ petlem via ~ disjlem18 , (general version of the former ~ prtlem17 ). (Contributed by Peter Mazsa, 10-Sep-2021.) $) disjlem17 $p |- ( Disj R -> ( ( x e. dom R /\ A e. [ x ] R ) -> ( E. y e. dom R ( A e. [ y ] R /\ B e. [ y ] R ) -> B e. [ x ] R ) ) ) $= ( wdisjALTV cv cdm wcel cec wa wrex wi df-rex an32 wceq disjlem14 biimprd wex eleq2 syl8 exp4a impd syl5bir expd imp5a imp4b exlimdv syl5bi ex ) EF ZAGZEHZIZCULEJZIZKZCBGZEJZIZDUSIZKZBUMLZDUOIZMVCURUMIZVBKZBSUKUQKZVDVBBUM NVGVFVDBUKUQVEVBVDUKUQVEUTVAVDUKUQVEUTVAVDMZMZUQVEKUNVEKZUPKUKVIUNVEUPOUK VJUPVIUKVJUPUTVHUKVJUPUTKUOUSPZVHABCEQVKVDVAUOUSDTRUAUBUCUDUEUFUGUHUIUJ $. $} ${ $d A x y $. $d B x y $. $d R x y $. $d V x y $. $d W x y $. $( Lemma for ~ disjdmqseq , ~ partim2 and ~ petlem via ~ disjlem19 , (general version of the former ~ prtlem18 ). (Contributed by Peter Mazsa, 16-Sep-2021.) $) disjlem18 $p |- ( ( A e. V /\ B e. W ) -> ( Disj R -> ( ( x e. dom R /\ A e. [ x ] R ) -> ( B e. [ x ] R <-> A ,~ R B ) ) ) ) $= ( vy wcel wa wdisjALTV cv cec wb wi wrex adantl relbrcoss impel sylibrd ex cdm ccoss wbr rspe expr wrel disjrel adantr disjlem17 imbi1d impbidd ) BEHCFHIZDJZAKZDUAZHZBUNDLZHZIZCUQHZBCDUBUCZMNULUMIZUSUTVAVBUSUTVANVBUSIUT URUTIZAUOOZVAUSUTVDNVBUPURUTVDVCAUOUDUEPVBVAVDMZUSULDUFZVEUMABCDEFQDUGZRU HSTVBUSBGKDLZHCVHHIGUOOZUTNZVAUTNUMUSVJNULAGBCDUIPVBVAVIUTULVFVAVIMUMGBCD EFQVGRUJSUKT $. $} ${ $d A x y $. $d A x z $. $d R x y $. $d R x z $. $d V x y $. $d V x z $. $( Lemma for ~ disjdmqseq , ~ partim2 and ~ petlem via ~ disjdmqs , (general version of the former ~ prtlem19 ). (Contributed by Peter Mazsa, 16-Sep-2021.) $) disjlem19 $p |- ( A e. V -> ( Disj R -> ( ( x e. dom R /\ A e. [ x ] R ) -> [ x ] R = [ A ] ,~ R ) ) ) $= ( vz wcel wdisjALTV cv cdm cec wa ccoss wceq wbr wb disjlem18 el2v2 imp31 wi cvv elecALTV ad2antrr bitr4d eqrdv exp31 ) BDFZCGZAHZCIFBUHCJZFKZUIBCL ZJZMUFUGKUJKZEUIULUMEHZUIFZBUNUKNZUNULFZUFUGUJUOUPOZUFUGUJURSSEABUNCDTPQR UFUQUPOZUGUJUFUSEBUNUKDTUAQUBUCUDUE $. $} ${ $d R u v x $. $( Lemma for ~ disjdmqseq via ~ disjdmqs . (Contributed by Peter Mazsa, 16-Sep-2021.) $) disjdmqsss $p |- ( Disj R -> ( dom R /. R ) C_ ( dom ,~ R /. ,~ R ) ) $= ( vv vx vu wdisjALTV cdm cqs cv wcel cec wceq wrex wa wb cvv elv syl wral wi reximi ccoss wrel disjrel releldmqs disjlem19 ralrimivv 2r19.29 sylbid ex ancom eqtr sylbi syl6 releldmqscoss sylibrd ssrdv ) AEZBAFZAGZAUAZFUTG ZUQBHZUSIZVBCHZUTJZKZCDHZAJZLZDURLZVBVAIZUQVCVHVEKZVBVHKZMZCVHLZDURLZVJUQ VCVMCVHLDURLZVPUQAUBZVCVQNZAUCZVRVSSBCDVBAOUDPQUQVLCVHRDURRZVQVPSUQVLDCUR VHUQVGURIVDVHIMVLSSCDVDAOUEPUFWAVQVPVLVMDCURVHUGUIQUHVOVIDURVNVFCVHVNVMVL MVFVLVMUJVBVHVEUKULTTUMUQVRVKVJNZVTVRWBSBCDVBAOUNPQUOUP $. $} ${ $d R u v x $. $( Lemma for ~ disjdmqseq via ~ disjdmqs . (Contributed by Peter Mazsa, 16-Sep-2021.) $) disjdmqscossss $p |- ( Disj R -> ( dom ,~ R /. ,~ R ) C_ ( dom R /. R ) ) $= ( vv vu vx cv cdm cqs wcel cab cec wceq wrex cvv elv syl wral reximi syl6 wa wi wdisjALTV wrel wb disjrel releldmqscoss disjlem19 ralrimivv 2r19.29 ccoss ex sylbid eqtr3 wex df-rex 19.41v bitri simprbi eqcom rexbii syl6ib ss2abdv abid2 eqcomi df-qs 3sstr4g ) AUAZBEZAUIZFVHGZHZBIZVGCEZAJZKZCAFZL ZBIVIVOAGVFVJVPBVFVJVMVGKZCVOLZVPVFVJVQDVMLZCVOLZVRVFVJVMDEZVHJZKZVGWBKZS ZDVMLZCVOLZVTVFVJWDDVMLCVOLZWGVFAUBZVJWHUCZAUDWIWJTBDCVGAMUENOVFWCDVMPCVO PZWHWGTVFWCCDVOVMVFVLVOHWAVMHZSWCTTDCWAAMUFNUGWKWHWGWCWDCDVOVMUHUJOUKWFVS CVOWEVQDVMVMVGWBULQQRVSVQCVOVSWLDUMZVQVSWLVQSDUMWMVQSVQDVMUNWLVQDUOUPUQQR VQVNCVOVMVGURUSUTVAVKVIBVIVBVCCBVOAVDVE $. $} $( If a relation is disjoint, its domain quotient is equal to the domain quotient of the cosets by it. Lemma for ~ partim2 and ~ petlem via ~ disjdmqseq . (Contributed by Peter Mazsa, 16-Sep-2021.) $) disjdmqs $p |- ( Disj R -> ( dom R /. R ) = ( dom ,~ R /. ,~ R ) ) $= ( wdisjALTV cdm cqs ccoss disjdmqsss disjdmqscossss eqssd ) ABACADAEZCIDAFA GH $. $( If a relation is disjoint, its domain quotient is equal to a class if and only if the domain quotient of the cosets by the relation is equal to the class. General version of ~ eldisjn0el (which is the closest theorem to the former ~ prter2 ). Lemma for ~ partim2 and ~ petlem . (Contributed by Peter Mazsa, 16-Sep-2021.) $) disjdmqseq $p |- ( Disj R -> ( ( dom R /. R ) = A <-> ( dom ,~ R /. ,~ R ) = A ) ) $= ( wdisjALTV cdm cqs ccoss disjdmqs eqeq1d ) BCBDBEBFZDIEABGH $. $( Special case of ~ disjdmqseq (perhaps this is the closest theorem to the former ~ prter2 ). (Contributed by Peter Mazsa, 26-Sep-2021.) $) eldisjn0el $p |- ( ElDisj A -> ( -. (/) e. A <-> ( U. A /. ~ A ) = A ) ) $= ( cep ccnv cres wdisjALTV cdm cqs wceq ccoss wb weldisj c0 wcel cuni ccoels wn disjdmqseq df-eldisj n0el3 dmqs1cosscnvepreseq bicomi bibi12i 3imtr4i ) BCADZEUDFUDGAHZUDIZFUFGAHZJAKLAMPZANAOGAHZJAUDQARUHUEUIUGASUGUIATUAUBUC $. $( Disjoint relation on its natural domain implies equivalence relation by the cosets of the relation, on its natural domain, cf. ~ partim . Lemma for ~ petlem . (Contributed by Peter Mazsa, 17-Sep-2021.) $) partim2 $p |- ( ( Disj R /\ ( dom R /. R ) = A ) -> ( EqvRel ,~ R /\ ( dom ,~ R /. ,~ R ) = A ) ) $= ( wdisjALTV cdm cqs wceq ccoss weqvrel disjim adantr disjdmqseq biimpa jca wa ) BCZBDBEAFZNBGZHZQDQEAFZORPBIJOPSABKLM $. $( Partition implies equivalence relation by the cosets of the relation on its natural domain, cf. ~ partim2 . (Contributed by Peter Mazsa, 17-Sep-2021.) $) partim $p |- ( R Part A -> ,~ R ErALTV A ) $= ( wdisjALTV cdm cqs wa ccoss weqvrel wpart werALTV partim2 dfpart2 dferALT2 wceq 3imtr4i ) BCBDBEANFBGZHPDPEANFABIAPJABKABLAPMO $. ${ $d A u $. $d B u $. $d V u $. $( Special case of ~ disjlem19 (together with ~ membpartlem19 , this is former ~ prtlem19 ). (Contributed by Peter Mazsa, 21-Oct-2021.) $) eldisjlem19 $p |- ( B e. V -> ( ElDisj A -> ( ( u e. dom ( `' _E |` A ) /\ B e. u ) -> u = [ B ] ~ A ) ) ) $= ( wcel weldisj cv cep ccnv cres cdm wa ccoels cec wceq wi ccoss wdisjALTV df-eldisj disjlem19 syl5bi imp expdimp wb eccnvepres3 eleq2d eqeq1d mpbid imbi12d adantl df-coels eceq2i eqeq2i syl6ibr expimpd ex ) CDEZBFZAGZHIBJ ZKEZCUSEZLUSCBMZNZOZPUQURLZVAVBVEVFVALZVBUSCUTQZNZOZVEVGCUSUTNZEZVKVIOZPZ VBVJPZVFVAVLVMUQURVAVLLVMPZURUTRUQVPBSACUTDTUAUBUCVAVNVOUDVFVAVLVBVMVJVAV KUSCBUSUEZUFVAVKUSVIVQUGUIUJUHVDVIUSVCVHCBUKULUMUNUOUP $. $} ${ $d A u $. $d B u $. $d V u $. $( Together with ~ disjlem19 , this is former ~ prtlem19 . (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 21-Oct-2021.) $) membpartlem19 $p |- ( B e. V -> ( MembPart A -> ( ( u e. A /\ B e. u ) -> u = [ B ] ~ A ) ) ) $= ( wcel wmembpart cv wa ccoels cec wceq wi weldisj c0 dfmembpart2 cep ccnv wn cres cdm n0el2 biimpi ad2antll eleq2d eldisjlem19 adantrd expd sylbird imp sylan2b impd ex ) CDEZBFZAGZBEZCUOEZHUOCBIJKZLUMUNHUPUQURUNUMBMZNBERZ HZUPUQURLZLBOUMVAHZUPUOPQBSTZEZVBVCVDBUOUTVDBKZUMUSUTVFBUAUBUCUDVCVEUQURU MVAVEUQHURLZUMUSVGUTABCDUEUFUIUGUHUJUKUL $. $} ${ petlem.1 $e |- ( ( EqvRel ,~ R /\ ( dom ,~ R /. ,~ R ) = A ) -> Disj R ) $. $( If you can prove that the equivalence of cosets on their natural domain implies disjointness (e.g. ~ eqvrelqseqdisj5 ), or converse function (cf. ~ dfdisjALTV ), then disjointness, and equivalence of cosets, both on their natural domain, are equivalent. Lemma for the Partition Equivalence Theorem ~ pet2 . (Contributed by Peter Mazsa, 18-Sep-2021.) $) petlem $p |- ( ( Disj R /\ ( dom R /. R ) = A ) <-> ( EqvRel ,~ R /\ ( dom ,~ R /. ,~ R ) = A ) ) $= ( wdisjALTV cdm cqs wa ccoss weqvrel partim2 simpr jca disjdmqseq pm5.32i wceq sylibr impbii ) BDZBEBFAOZGZBHZIZUAEUAFAOZGZABJUDRUCGTUDRUCCUBUCKLRS UCABMNPQ $. $} ${ petlemi.1 $e |- Disj R $. $( If you can prove disjointness (e.g. ~ disjALTV0 , ~ disjALTVid , ~ disjALTVidres , ~ disjALTVtxpidres , search for theorems containing the ' |- Disj ' string), or the same with converse function (cf. ~ dfdisjALTV ), then disjointness, and equivalence of cosets, both on their natural domain, are equivalent. Inference version. (Contributed by Peter Mazsa, 18-Sep-2021.) $) petlemi $p |- ( ( Disj R /\ ( dom R /. R ) = A ) <-> ( EqvRel ,~ R /\ ( dom ,~ R /. ,~ R ) = A ) ) $= ( wdisjALTV ccoss weqvrel cdm cqs wceq wa a1i petlem ) ABBDBEZFMGMHAIJCKL $. $} ${ petlemd.1 $e |- ( ph -> Disj R ) $. $( If you can prove disjointness (or converse function, cf. ~ dfdisjALTV ), then disjointness, and equivalence of cosets, both on their natural domain, are equivalent. Deduction version. (Contributed by Peter Mazsa, 18-Sep-2021.) $) petlemd $p |- ( ph -> ( ( Disj R /\ ( dom R /. R ) = A ) <-> ( EqvRel ,~ R /\ ( dom ,~ R /. ,~ R ) = A ) ) ) $= ( wdisjALTV cdm cqs wa ccoss weqvrel partim2 a1d simpr disjdmqseq pm5.32i wceq jca2 syl6ibr impbid2 ) ACEZCFCGBPZHZCIZJZUCFUCGBPZHZBCKAUFTUEHUBAUFT UEATUFDLUDUEMQTUAUEBCNORS $. $} $( Class ` A ` is a partition by the null class if and only if the cosets by the null class is an equivalence relation on it. (Contributed by Peter Mazsa, 31-Dec-2021.) $) pet02 $p |- ( ( Disj (/) /\ ( dom (/) /. (/) ) = A ) <-> ( EqvRel ,~ (/) /\ ( dom ,~ (/) /. ,~ (/) ) = A ) ) $= ( c0 disjALTV0 petlemi ) ABCD $. $( A class is a partition by the null class if and only if the cosets by the null class is an equivalence relation on it. (Contributed by Peter Mazsa, 31-Dec-2021.) $) pet0 $p |- ( (/) Part A <-> ,~ (/) ErALTV A ) $= ( c0 wdisjALTV cdm cqs wceq wa ccoss weqvrel wpart werALTV dfpart2 dferALT2 pet02 3bitr4i ) BCBDBEAFGBHZIPDPEAFGABJAPKANABLAPMO $. $( Class ` A ` is a partition by the class of the identity relations if and only if the cosets by the identity class is an equivalence relation on it. (Contributed by Peter Mazsa, 31-Dec-2021.) $) petid2 $p |- ( ( Disj _I /\ ( dom _I /. _I ) = A ) <-> ( EqvRel ,~ _I /\ ( dom ,~ _I /. ,~ _I ) = A ) ) $= ( cid disjALTVid petlemi ) ABCD $. $( A class is a partition by the class of the identity relations if and only if the cosets by the identity class is an equivalence relation on it. (Contributed by Peter Mazsa, 31-Dec-2021.) $) petid $p |- ( _I Part A <-> ,~ _I ErALTV A ) $= ( cid wdisjALTV cdm cqs wceq wa ccoss weqvrel wpart werALTV petid2 dferALT2 dfpart2 3bitr4i ) BCBDBEAFGBHZIPDPEAFGABJAPKALABNAPMO $. $( Class ` A ` is a partition by the class of the identity relations restricted to it if and only if the cosets by the restricted identity class is an equivalence relation on it. (Contributed by Peter Mazsa, 31-Dec-2021.) $) petidres2 $p |- ( ( Disj ( _I |` A ) /\ ( dom ( _I |` A ) /. ( _I |` A ) ) = A ) <-> ( EqvRel ,~ ( _I |` A ) /\ ( dom ,~ ( _I |` A ) /. ,~ ( _I |` A ) ) = A ) ) $= ( cid cres disjALTVidres petlemi ) ABACADE $. $( A class is a partition by the class of the identity relations restricted to it if and only if the cosets by the restricted identity class is an equivalence relation on it, cf. ~ eqvrel1cossidres . (Contributed by Peter Mazsa, 31-Dec-2021.) $) petidres $p |- ( ( _I |` A ) Part A <-> ,~ ( _I |` A ) ErALTV A ) $= ( cid cres wdisjALTV cdm wceq ccoss weqvrel wpart werALTV petidres2 dfpart2 cqs wa dferALT2 3bitr4i ) BACZDQEQMAFNQGZHRERMAFNAQIARJAKAQLAROP $. $( Class ` A ` is a partition by an intersection with the class of the identity relations restricted to it if and only if the cosets by the intersection is an equivalence relation on it. (Contributed by Peter Mazsa, 31-Dec-2021.) $) petinidres2 $p |- ( ( Disj ( R i^i ( _I |` A ) ) /\ ( dom ( R i^i ( _I |` A ) ) /. ( R i^i ( _I |` A ) ) ) = A ) <-> ( EqvRel ,~ ( R i^i ( _I |` A ) ) /\ ( dom ,~ ( R i^i ( _I |` A ) ) /. ,~ ( R i^i ( _I |` A ) ) ) = A ) ) $= ( cid cres cin disjALTVinidres petlemi ) ABCADEABFG $. $( A class is a partition by an intersection with the class of the identity relations restricted to it if and only if the cosets by the intersection is an equivalence relation on it. Cf. ~ br1cossinidres , ~ disjALTVinidres and ~ eqvrel1cossinidres . (Contributed by Peter Mazsa, 31-Dec-2021.) $) petinidres $p |- ( ( R i^i ( _I |` A ) ) Part A <-> ,~ ( R i^i ( _I |` A ) ) ErALTV A ) $= ( cid cres cin wdisjALTV cdm cqs wa ccoss weqvrel wpart werALTV petinidres2 wceq dfpart2 dferALT2 3bitr4i ) BCADEZFSGSHAOISJZKTGTHAOIASLATMABNASPATQR $. $( Class ` A ` is a partition by a tail Cartesian product with the class of the identity relations restricted to it if and only if the cosets by the tail Cartesian product is an equivalence relation on it. (Contributed by Peter Mazsa, 31-Dec-2021.) $) pettxpidres2 $p |- ( ( Disj ( R (x) ( _I |` A ) ) /\ ( dom ( R (x) ( _I |` A ) ) /. ( R (x) ( _I |` A ) ) ) = A ) <-> ( EqvRel ,~ ( R (x) ( _I |` A ) ) /\ ( dom ,~ ( R (x) ( _I |` A ) ) /. ,~ ( R (x) ( _I |` A ) ) ) = A ) ) $= ( cid cres ctxp disjALTVtxpidres petlemi ) ABCADEABFG $. $( A class is a partition by a tail Cartesian product with the class of the identity relations restricted to it if and only if the cosets by the tail Cartesian product is an equivalence relation on it. Cf. ~ br1cosstxpidres , ~ disjALTVtxpidres and ~ eqvrel1cosstxpidres . (Contributed by Peter Mazsa, 31-Dec-2021.) $) pettxpidres $p |- ( ( R (x) ( _I |` A ) ) Part A <-> ,~ ( R (x) ( _I |` A ) ) ErALTV A ) $= ( cid cres ctxp wdisjALTV cdm wceq ccoss weqvrel wpart werALTV pettxpidres2 cqs wa dfpart2 dferALT2 3bitr4i ) BCADEZFSGSNAHOSIZJTGTNAHOASKATLABMASPATQR $. ${ $d A x y $. $d R x y $. $( The elements of the quotient set of an equivalence relation are disjoint (cf. ~ eqvreldisj3 ). (Contributed by Mario Carneiro, 10-Dec-2016.) (Revised by Peter Mazsa, 19-Sep-2021.) $) eqvreldisj2 $p |- ( EqvRel R -> ElDisj ( A /. R ) ) $= ( vx vy weqvrel cv wceq cin c0 wo cqs wral weldisj wa simpl simprl simprr wcel qsdisjALTV ralrimivva dfeldisj5 sylibr ) BEZCFZDFZGUDUEHIGJZDABKZLCU GLUGMUCUFCDUGUGUCUDUGRZUEUGRZNZNAUDUEBUCUJOUCUHUIPUCUHUIQSTDCUGUAUB $. $} $( The elements of the quotient set of an equivalence relation are disjoint (cf. ~ qsdisj2 ). (Contributed by Mario Carneiro, 10-Dec-2016.) (Revised by Peter Mazsa, 20-Jun-2019.) (Revised by Peter Mazsa, 19-Sep-2021.) $) eqvreldisj3 $p |- ( EqvRel R -> Disj ( `' _E |` ( A /. R ) ) ) $= ( weqvrel cqs weldisj cep ccnv cres wdisjALTV eqvreldisj2 df-eldisj sylib ) BCABDZEFGMHIABJMKL $. $( Intersection with the elements of the quotient set of an equivalence relation is disjoint. (Contributed by Peter Mazsa, 30-May-2020.) (Revised by Peter Mazsa, 31-Dec-2021.) $) eqvreldisj4 $p |- ( EqvRel R -> Disj ( S i^i ( `' _E |` ( B /. R ) ) ) ) $= ( weqvrel cep ccnv cqs cres wdisjALTV cin eqvreldisj3 disjimin syl ) BDEFAB GHZICNJIABKCNLM $. $( Tail Cartesian product with the elements of the quotient set of an equivalence relation is disjoint. (Contributed by Peter Mazsa, 30-May-2020.) (Revised by Peter Mazsa, 22-Sep-2021.) $) eqvreldisj5 $p |- ( EqvRel R -> Disj ( S (x) ( `' _E |` ( B /. R ) ) ) ) $= ( weqvrel cep ccnv cqs cres wdisjALTV ctxp eqvreldisj3 disjimtxp syl ) BDEF ABGHZICNJIABKCNLM $. $( Implication of ~ eqvreldisj2 , lemma for The Main Theorem of Equivalences ~ mainer . (Contributed by Peter Mazsa, 23-Sep-2021.) $) eqvrelqseqdisj2 $p |- ( ( EqvRel R /\ ( B /. R ) = A ) -> ElDisj A ) $= ( weqvrel cqs wceq wa weldisj eqvreldisj2 adantr wb eldisjeq adantl mpbid ) CDZBCEZAFZGPHZAHZORQBCIJQRSKOPALMN $. $( Implication of ~ eqvreldisj3 , lemma for the Membership Partition Equivalence Theorem ~ mpet3 . (Contributed by Peter Mazsa, 27-Oct-2020.) (Revised by Peter Mazsa, 24-Sep-2021.) $) eqvrelqseqdisj3 $p |- ( ( EqvRel R /\ ( B /. R ) = A ) -> Disj ( `' _E |` A ) ) $= ( weqvrel cqs wceq wa cep ccnv cres wdisjALTV eqvreldisj3 adantr wb disjeqd reseq2 adantl mpbid ) CDZBCEZAFZGHIZTJZKZUBAJZKZSUDUABCLMUAUDUFNSUAUCUETAUB POQR $. $( Lemma for ~ petincnvepres2 . (Contributed by Peter Mazsa, 31-Dec-2021.) $) eqvrelqseqdisj4 $p |- ( ( EqvRel R /\ ( B /. R ) = A ) -> Disj ( S i^i ( `' _E |` A ) ) ) $= ( weqvrel cqs wceq cep ccnv cres wdisjALTV cin eqvrelqseqdisj3 disjimin syl wa ) CEBCFAGPHIAJZKDQLKABCMDQNO $. $( Lemma for the Partition-Equivalence Theorem ~ pet2 . (Contributed by Peter Mazsa, 15-Jul-2020.) (Revised by Peter Mazsa, 22-Sep-2021.) $) eqvrelqseqdisj5 $p |- ( ( EqvRel R /\ ( B /. R ) = A ) -> Disj ( S (x) ( `' _E |` A ) ) ) $= ( weqvrel cqs wceq wa cep ccnv cres wdisjALTV eqvrelqseqdisj3 disjimtxp syl ctxp ) CEBCFAGHIJAKZLDQPLABCMDQNO $. $( The Main Theorem of Equivalences: any equivalence relation implies equivalent membership as well. (Contributed by Peter Mazsa, 26-Sep-2021.) $) mainer $p |- ( R ErALTV A -> MembEr A ) $= ( weqvrel cdm cqs wa ccoels werALTV wmember weldisj eqvrelqseqdisj2 mdisjim wceq cuni syl c0 wcel wn n0eldmqseq adantl wb eldisjn0el mpbid jca dferALT2 dfmember3 3imtr4i ) BCZBDZBEAMZFZAGZCZANULEAMZFABHAIUKUMUNUKAJZUMAUIBKZALOU KPAQRZUNUJUQUHABSTUKUOUQUNUAUPAUBOUCUDABUEAUFUG $. $( Membership Partition-Equivalence Theorem. Together with ~ mpet and ~ mpet2 , this is what is generally identified as the partition equivalence theorem (but cf. ~ pet2 with general ` R ` ). (Contributed by Peter Mazsa, 4-May-2018.) (Revised by Peter Mazsa, 26-Sep-2021.) $) mpet3 $p |- ( ( ElDisj A /\ -. (/) e. A ) <-> ( EqvRel ~ A /\ ( U. A /. ~ A ) = A ) ) $= ( weldisj c0 wcel wn wa cep ccnv cres wdisjALTV cdm cqs wceq weqvrel ccoels ccoss cuni df-eldisj n0el3 anbi12i eqvrelqseqdisj3 petlem eqvreldmqscoels 3bitri ) ABZCADEZFGHAIZJZUGKUGLAMZFUGPZNUJKZUJLAMFAOZNAQULLAMFUEUHUFUIARAST AUGAUKUJUAUBAUCUD $. $( Membership Partition-Equivalence Theorem in its shortest possible form: membership equivalence relation and membership partition are the same (or: each element of ` A ` has equivalent members if and only if ` A ` is a membership partition). Together with ~ mpet2 and ~ mpet3 , this is what is generally identified as the partition-equivalence theorem (but cf. ~ pet with general ` R ` ). (Contributed by Peter Mazsa, 24-Sep-2021.) $) mpet $p |- ( MembPart A <-> MembEr A ) $= ( weldisj c0 wcel wn ccoels weqvrel cuni wceq wmembpart wmember dfmembpart2 wa cqs mpet3 dfmember3 3bitr4i ) ABCADEMAFZGAHRNAIMAJAKAOALAPQ $. $( Membership Partition-Equivalence Theorem in a shorter form. Together with ~ mpet and ~ mpet3 , this is what is generally identified as the partition-equivalence theorem (but cf. ~ pet with general ` R ` ). (Contributed by Peter Mazsa, 24-Sep-2021.) $) mpet2 $p |- ( ( `' _E |` A ) Part A <-> ,~ ( `' _E |` A ) ErALTV A ) $= ( wmembpart wmember cep ccnv cres wpart ccoss werALTV df-membpart dfmember2 mpet 3bitr3i ) ABACADEAFZGANHIALAJAKM $. $( Membership Partition-Equivalence Theorem with binary relations, cf. ~ mpet2 . (Contributed by Peter Mazsa, 24-Sep-2021.) $) mpets $p |- ( A e. V -> ( ( `' _E |` A ) Parts A <-> ,~ ( `' _E |` A ) Ers A ) ) $= ( wcel cep ccnv cres cparts wbr ccoss wb wpart werALTV mpet2 cvv cnvepresex cers brpartspart mpdan 1cosscnvepresex brerserALTV bibi12d mpbiri ) ABCZDEA FZAGHZUDIZAPHZJAUDKZAUFLZJAMUCUEUHUGUIUCUDNCUEUHJABOAUDBNQRUCUFNCUGUIJABSAU FBNTRUAUB $. $( The Theorem of Fences by Equivalences: all kinds of equivalence relations imaginable (in addition to the membership equivalence relation cf. ~ mpet ) generate partition of the members. (Contributed by Peter Mazsa, 26-Sep-2021.) $) fences $p |- ( R ErALTV A -> MembPart A ) $= ( werALTV wmember wmembpart mainer mpet sylibr ) ABCADAEABFAGH $. $( The Theorem of Fences by Equivalences: all kinds of equivalence relations imaginable (in addition to the membership equivalence relation cf. ~ mpet3 ) imply the partition of the members, moreover it implies that ` ( R ErALTV A -> ElDisj A ) ` and that ` ( R ErALTV A -> -. (/) e. A ) ` . (Contributed by Peter Mazsa, 15-Oct-2021.) $) fences2 $p |- ( R ErALTV A -> ( ElDisj A /\ -. (/) e. A ) ) $= ( werALTV wmembpart weldisj c0 wcel wn wa fences dfmembpart2 sylib ) ABCADA EFAGHIABJAKL $. $( The Main Theorem of Equivalences: any equivalence relation implies equivalent membership as well. (Contributed by Peter Mazsa, 15-Oct-2021.) $) mainer2 $p |- ( R ErALTV A -> ( EqvRel ~ A /\ -. (/) e. A ) ) $= ( werALTV weldisj c0 wcel wn wa ccoels weqvrel fences2 mdisjim anim1i syl ) ABCADZEAFGZHAIJZPHABKOQPALMN $. $( Any equivalence relation implies equivalent coelements as well. (Contributed by Peter Mazsa, 20-Oct-2021.) $) mainerim $p |- ( R ErALTV A -> EqvRel ~ A ) $= ( werALTV ccoels weqvrel c0 wcel wn mainer2 simpld ) ABCADEFAGHABIJ $. $( A partition-equivalence theorem with intersection and general ` R ` . (Contributed by Peter Mazsa, 31-Dec-2021.) $) petincnvepres2 $p |- ( ( Disj ( R i^i ( `' _E |` A ) ) /\ ( dom ( R i^i ( `' _E |` A ) ) /. ( R i^i ( `' _E |` A ) ) ) = A ) <-> ( EqvRel ,~ ( R i^i ( `' _E |` A ) ) /\ ( dom ,~ ( R i^i ( `' _E |` A ) ) /. ,~ ( R i^i ( `' _E |` A ) ) ) = A ) ) $= ( cep ccnv cres cin ccoss cdm eqvrelqseqdisj4 petlem ) ABCDAEFZAKGZHLBIJ $. $( The shortest form of a partition-equivalence theorem with intersection and general ` R ` . Cf. ~ br1cossincnvepres . (Contributed by Peter Mazsa, 23-Sep-2021.) $) petincnvepres $p |- ( ( R i^i ( `' _E |` A ) ) Part A <-> ,~ ( R i^i ( `' _E |` A ) ) ErALTV A ) $= ( cep ccnv cres cin wdisjALTV cdm cqs wa ccoss weqvrel wpart petincnvepres2 wceq werALTV dfpart2 dferALT2 3bitr4i ) BCDAEFZGTHTIAOJTKZLUAHUAIAOJATMAUAP ABNATQAUARS $. $( Partition-Equivalence Theorem, with general ` R ` . This theorem (together with ~ pet and ~ pets ) is the main result of my inquiry into set theory, cf. the comment of ~ pet . (Contributed by Peter Mazsa, 24-May-2021.) (Revised by Peter Mazsa, 23-Sep-2021.) $) pet2 $p |- ( ( Disj ( R (x) ( `' _E |` A ) ) /\ ( dom ( R (x) ( `' _E |` A ) ) /. ( R (x) ( `' _E |` A ) ) ) = A ) <-> ( EqvRel ,~ ( R (x) ( `' _E |` A ) ) /\ ( dom ,~ ( R (x) ( `' _E |` A ) ) /. ,~ ( R (x) ( `' _E |` A ) ) ) = A ) ) $= ( cep ccnv cres ctxp ccoss cdm eqvrelqseqdisj5 petlem ) ABCDAEFZAKGZHLBIJ $. $( Partition-Equivalence Theorem with general ` R ` while keeping the restricted converse epsilon relation of ~ mpet2 (as opposed to ~ petincnvepres ). Cf. ~ br1cosstxpcnvepres . This theorem (together with ~ pets and ~ pet2 ) is the main result of my inquiry into set theory. Not more general than the conventional Membership Partition-Equivalence Theorem ~ mpet , ~ mpet2 and ~ mpet3 (because you cannot set ` R ` in this theorem in a way that you would get ~ mpet2 ), i.e., this is not the hypothetical General Partition Equivalence Theorem gpet ` |- ( R Part A <-> ,~ R ErALTV A ) ` , but this has a general part that ~ mpet2 lacks: ` R ` , which is enough for my future application of set theory, for my purpose outside of set theory. Motto: "Le meglio è l'inimico del bene." (Contributed by Peter Mazsa, 23-Sep-2021.) $) pet $p |- ( ( R (x) ( `' _E |` A ) ) Part A <-> ,~ ( R (x) ( `' _E |` A ) ) ErALTV A ) $= ( cep ccnv cres ctxp wdisjALTV cdm wceq wa ccoss weqvrel wpart werALTV pet2 cqs dfpart2 dferALT2 3bitr4i ) BCDAEFZGTHTPAIJTKZLUAHUAPAIJATMAUANABOATQAUA RS $. $( Some kinds of partition with general ` R ` (in addition to the membership partition cf. ~ mpet and ~ mpet2 ) imply equivalence of members. (Contributed by Peter Mazsa, 23-Sep-2021.) $) partimmember $p |- ( ( R (x) ( `' _E |` A ) ) Part A -> MembEr A ) $= ( cep ccnv cres ctxp wpart ccoss werALTV wmember pet mainer sylbi ) ABCDAEF ZGANHZIAJABKAOLM $. $( Some kinds of partition with general ` R ` imply membership partition as well. (Contributed by Peter Mazsa, 23-Sep-2021.) $) partimmembpart $p |- ( ( R (x) ( `' _E |` A ) ) Part A -> MembPart A ) $= ( cep ccnv cres ctxp wpart wmember wmembpart partimmember mpet sylibr ) ABC DAEFGAHAIABJAKL $. $( Partition-Equivalence Theorem with general ` R ` , with binary relations. This theorem (together with ~ pet and ~ pet2 ) is the main result of my inquiry into set theory, cf. the comment of ~ pet . (Contributed by Peter Mazsa, 23-Sep-2021.) $) pets $p |- ( ( A e. V /\ R e. W ) -> ( ( R (x) ( `' _E |` A ) ) Parts A <-> ,~ ( R (x) ( `' _E |` A ) ) Ers A ) ) $= ( wcel wa cep ccnv cres ctxp cparts wbr ccoss cers wb wpart werALTV pet cvv syl2anc simpl txpcnvepresex brpartspart 1cosstxpcnvepresex bibi12d mpbiri brerserALTV ) ACEZBDEZFZBGHAIJZAKLZUKMZANLZOAUKPZAUMQZOABRUJULUOUNUPUJUHUKS EULUOOUHUIUAZABCDUBAUKCSUCTUJUHUMSEUNUPOUQABCDUDAUMCSUGTUEUF $. $( (End of Peter Mazsa's mathbox.) $)