P. 69 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6801-6900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ercnv 6801	The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.)
|- ( R Er A -> `' R = R )
Theorem	errn 6802	The range and domain of an equivalence relation are equal. (Contributed by Rodolfo Medina, 11-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
|- ( R Er A -> ran R = A )
Theorem	erssxp 6803	An equivalence relation is a subset of the cartesian product of the field. (Contributed by Mario Carneiro, 12-Aug-2015.)
|- ( R Er A -> R C_ ( A X. A ) )
Theorem	erex 6804	An equivalence relation is a set if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)
|- ( R Er A -> ( A e. V -> R e. _V ) )
Theorem	erexb 6805	An equivalence relation is a set if and only if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
|- ( R Er A -> ( R e. _V <-> A e. _V ) )
Theorem	iserd 6806*	A reflexive, symmetric, transitive relation is an equivalence relation on its domain. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
|- ( ph -> Rel R )   &    |- ( ( ph /\ x R y ) -> y R x )   &    |- ( ( ph /\ ( x R y /\ y R z ) ) -> x R z )   &    |- ( ph -> ( x e. A <-> x R x ) )   =>    |- ( ph -> R Er A )
Theorem	brdifun 6807	Evaluate the incomparability relation. (Contributed by Mario Carneiro, 9-Jul-2014.)
|- R = ( ( X X. X ) \ ( .< u. `' .< ) )   =>    |- ( ( A e. X /\ B e. X ) -> ( A R B <-> -. ( A .< B \/ B .< A ) ) )
Theorem	swoer 6808*	Incomparability under a strict weak partial order is an equivalence relation. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
|- R = ( ( X X. X ) \ ( .< u. `' .< ) )   &    |- ( ( ph /\ ( y e. X /\ z e. X ) ) -> ( y .< z -> -. z .< y ) )   &    |- ( ( ph /\ ( x e. X /\ y e. X /\ z e. X ) ) -> ( x .< y -> ( x .< z \/ z .< y ) ) )   =>    |- ( ph -> R Er X )
Theorem	swoord1 6809*	The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.)
|- R = ( ( X X. X ) \ ( .< u. `' .< ) )   &    |- ( ( ph /\ ( y e. X /\ z e. X ) ) -> ( y .< z -> -. z .< y ) )   &    |- ( ( ph /\ ( x e. X /\ y e. X /\ z e. X ) ) -> ( x .< y -> ( x .< z \/ z .< y ) ) )   &    |- ( ph -> B e. X )   &    |- ( ph -> C e. X )   &    |- ( ph -> A R B )   =>    |- ( ph -> ( A .< C <-> B .< C ) )
Theorem	swoord2 6810*	The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.)
|- R = ( ( X X. X ) \ ( .< u. `' .< ) )   &    |- ( ( ph /\ ( y e. X /\ z e. X ) ) -> ( y .< z -> -. z .< y ) )   &    |- ( ( ph /\ ( x e. X /\ y e. X /\ z e. X ) ) -> ( x .< y -> ( x .< z \/ z .< y ) ) )   &    |- ( ph -> B e. X )   &    |- ( ph -> C e. X )   &    |- ( ph -> A R B )   =>    |- ( ph -> ( C .< A <-> C .< B ) )
Theorem	eqerlem 6811*	Lemma for eqer 6812. (Contributed by NM, 17-Mar-2008.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
|- ( x = y -> A = B )   &    |- R = { <. x , y >. | A = B }   =>    |- ( z R w <-> [_ z / x ]_ A = [_ w / x ]_ A )
Theorem	eqer 6812*	Equivalence relation involving equality of dependent classes A ( x ) and B ( y ). (Contributed by NM, 17-Mar-2008.) (Revised by Mario Carneiro, 12-Aug-2015.)
|- ( x = y -> A = B )   &    |- R = { <. x , y >. | A = B }   =>    |- R Er _V
Theorem	ider 6813	The identity relation is an equivalence relation. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 9-Jul-2014.)
|- _I Er _V
Theorem	0er 6814	The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.)
|- (/) Er (/)
Theorem	eceq1 6815	Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
|- ( A = B -> [ A ] C = [ B ] C )
Theorem	eceq1d 6816	Equality theorem for equivalence class (deduction form). (Contributed by Jim Kingdon, 31-Dec-2019.)
|- ( ph -> A = B )   =>    |- ( ph -> [ A ] C = [ B ] C )
Theorem	eceq2 6817	Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
|- ( A = B -> [ C ] A = [ C ] B )
Theorem	eceq2i 6818	Equality theorem for the A-coset and B-coset of C, inference version. (Contributed by Peter Mazsa, 11-May-2021.)
|- A = B   =>    |- [ C ] A = [ C ] B
Theorem	eceq2d 6819	Equality theorem for the A-coset and B-coset of C, deduction version. (Contributed by Peter Mazsa, 23-Apr-2021.)
|- ( ph -> A = B )   =>    |- ( ph -> [ C ] A = [ C ] B )
Theorem	elecg 6820	Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by Mario Carneiro, 9-Jul-2014.)
|- ( ( A e. V /\ B e. W ) -> ( A e. [ B ] R <-> B R A ) )
Theorem	elec 6821	Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.)
|- A e. _V   &    |- B e. _V   =>    |- ( A e. [ B ] R <-> B R A )
Theorem	relelec 6822	Membership in an equivalence class when R is a relation. (Contributed by Mario Carneiro, 11-Sep-2015.)
|- ( Rel R -> ( A e. [ B ] R <-> B R A ) )
Theorem	ecss 6823	An equivalence class is a subset of the domain. (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
|- ( ph -> R Er X )   =>    |- ( ph -> [ A ] R C_ X )
Theorem	ecdmn0m 6824*	A representative of an inhabited equivalence class belongs to the domain of the equivalence relation. (Contributed by Jim Kingdon, 21-Aug-2019.)
|- ( A e. dom R <-> E. x x e. [ A ] R )
Theorem	ereldm 6825	Equality of equivalence classes implies equivalence of domain membership. (Contributed by NM, 28-Jan-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)
|- ( ph -> R Er X )   &    |- ( ph -> [ A ] R = [ B ] R )   =>    |- ( ph -> ( A e. X <-> B e. X ) )
Theorem	erth 6826	Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)
|- ( ph -> R Er X )   &    |- ( ph -> A e. X )   =>    |- ( ph -> ( A R B <-> [ A ] R = [ B ] R ) )
Theorem	erth2 6827	Basic property of equivalence relations. Compare Theorem 73 of [Suppes] p. 82. Assumes membership of the second argument in the domain. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)
|- ( ph -> R Er X )   &    |- ( ph -> B e. X )   =>    |- ( ph -> ( A R B <-> [ A ] R = [ B ] R ) )
Theorem	erthi 6828	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.)
|- ( ph -> R Er X )   &    |- ( ph -> A R B )   =>    |- ( ph -> [ A ] R = [ B ] R )
Theorem	ecidsn 6829	An equivalence class modulo the identity relation is a singleton. (Contributed by NM, 24-Oct-2004.)
|- [ A ] _I = { A }
Theorem	qseq1 6830	Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
|- ( A = B -> ( A /. C ) = ( B /. C ) )
Theorem	qseq2 6831	Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
|- ( A = B -> ( C /. A ) = ( C /. B ) )
Theorem	elqsg 6832*	Closed form of elqs 6833. (Contributed by Rodolfo Medina, 12-Oct-2010.)
|- ( B e. V -> ( B e. ( A /. R ) <-> E. x e. A B = [ x ] R ) )
Theorem	elqs 6833*	Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
|- B e. _V   =>    |- ( B e. ( A /. R ) <-> E. x e. A B = [ x ] R )
Theorem	elqsi 6834*	Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
|- ( B e. ( A /. R ) -> E. x e. A B = [ x ] R )
Theorem	ecelqsg 6835	Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)
|- ( ( R e. V /\ B e. A ) -> [ B ] R e. ( A /. R ) )
Theorem	ecelqsi 6836	Membership of an equivalence class in a quotient set. (Contributed by NM, 25-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
|- R e. _V   =>    |- ( B e. A -> [ B ] R e. ( A /. R ) )
Theorem	ecopqsi 6837	"Closure" law for equivalence class of ordered pairs. (Contributed by NM, 25-Mar-1996.)
|- R e. _V   &    |- S = ( ( A X. A ) /. R )   =>    |- ( ( B e. A /\ C e. A ) -> [ <. B , C >. ] R e. S )
Theorem	qsexg 6838	A quotient set exists. (Contributed by FL, 19-May-2007.) (Revised by Mario Carneiro, 9-Jul-2014.)
|- ( A e. V -> ( A /. R ) e. _V )
Theorem	qsex 6839	A quotient set exists. (Contributed by NM, 14-Aug-1995.)
|- A e. _V   =>    |- ( A /. R ) e. _V
Theorem	uniqs 6840	The union of a quotient set. (Contributed by NM, 9-Dec-2008.)
|- ( R e. V -> U. ( A /. R ) = ( R " A ) )
Theorem	qsss 6841	A quotient set is a set of subsets of the base set. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
|- ( ph -> R Er A )   =>    |- ( ph -> ( A /. R ) C_ ~P A )
Theorem	uniqs2 6842	The union of a quotient set. (Contributed by Mario Carneiro, 11-Jul-2014.)
|- ( ph -> R Er A )   &    |- ( ph -> R e. V )   =>    |- ( ph -> U. ( A /. R ) = A )
Theorem	snec 6843	The singleton of an equivalence class. (Contributed by NM, 29-Jan-1999.) (Revised by Mario Carneiro, 9-Jul-2014.)
|- A e. _V   =>    |- { [ A ] R } = ( { A } /. R )
Theorem	ecqs 6844	Equivalence class in terms of quotient set. (Contributed by NM, 29-Jan-1999.)
|- R e. _V   =>    |- [ A ] R = U. ( { A } /. R )
Theorem	ecid 6845	A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
|- A e. _V   =>    |- [ A ] `' _E = A
Theorem	ecidg 6846	A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by Jim Kingdon, 8-Jan-2020.)
|- ( A e. V -> [ A ] `' _E = A )
Theorem	qsid 6847	A set is equal to its quotient set mod converse epsilon. (Note: converse epsilon is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
|- ( A /. `' _E ) = A
Theorem	ectocld 6848*	Implicit substitution of class for equivalence class. (Contributed by Mario Carneiro, 9-Jul-2014.)
|- S = ( B /. R )   &    |- ( [ x ] R = A -> ( ph <-> ps ) )   &    |- ( ( ch /\ x e. B ) -> ph )   =>    |- ( ( ch /\ A e. S ) -> ps )
Theorem	ectocl 6849*	Implicit substitution of class for equivalence class. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
|- S = ( B /. R )   &    |- ( [ x ] R = A -> ( ph <-> ps ) )   &    |- ( x e. B -> ph )   =>    |- ( A e. S -> ps )
Theorem	elqsn0m 6850*	An element of a quotient set is inhabited. (Contributed by Jim Kingdon, 21-Aug-2019.)
|- ( ( dom R = A /\ B e. ( A /. R ) ) -> E. x x e. B )
Theorem	elqsn0 6851	A quotient set doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)
|- ( ( dom R = A /\ B e. ( A /. R ) ) -> B =/= (/) )
Theorem	ecelqsdm 6852	Membership of an equivalence class in a quotient set. (Contributed by NM, 30-Jul-1995.)
|- ( ( dom R = A /\ [ B ] R e. ( A /. R ) ) -> B e. A )
Theorem	xpider 6853	A square Cartesian product is an equivalence relation (in general it's not a poset). (Contributed by FL, 31-Jul-2009.) (Revised by Mario Carneiro, 12-Aug-2015.)
|- ( A X. A ) Er A
Theorem	iinerm 6854*	The intersection of a nonempty family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)
|- ( ( E. y y e. A /\ A. x e. A R Er B ) -> |^|_ x e. A R Er B )
Theorem	riinerm 6855*	The relative intersection of a family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)
|- ( ( E. y y e. A /\ A. x e. A R Er B ) -> ( ( B X. B ) i^i |^|_ x e. A R ) Er B )
Theorem	erinxp 6856	A restricted equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 10-Jul-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
|- ( ph -> R Er A )   &    |- ( ph -> B C_ A )   =>    |- ( ph -> ( R i^i ( B X. B ) ) Er B )
Theorem	ecinxp 6857	Restrict the relation in an equivalence class to a base set. (Contributed by Mario Carneiro, 10-Jul-2015.)
|- ( ( ( R " A ) C_ A /\ B e. A ) -> [ B ] R = [ B ] ( R i^i ( A X. A ) ) )
Theorem	qsinxp 6858	Restrict the equivalence relation in a quotient set to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
|- ( ( R " A ) C_ A -> ( A /. R ) = ( A /. ( R i^i ( A X. A ) ) ) )
Theorem	qsel 6859	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.)
|- ( ( R Er X /\ B e. ( A /. R ) /\ C e. B ) -> B = [ C ] R )
Theorem	qliftlem 6860*	F, a function lift, is a subset of R X. S. (Contributed by Mario Carneiro, 23-Dec-2016.)
|- F = ran ( x e. X |-> <. [ x ] R , A >. )   &    |- ( ( ph /\ x e. X ) -> A e. Y )   &    |- ( ph -> R Er X )   &    |- ( ph -> X e. _V )   =>    |- ( ( ph /\ x e. X ) -> [ x ] R e. ( X /. R ) )
Theorem	qliftrel 6861*	F, a function lift, is a subset of R X. S. (Contributed by Mario Carneiro, 23-Dec-2016.)
|- F = ran ( x e. X |-> <. [ x ] R , A >. )   &    |- ( ( ph /\ x e. X ) -> A e. Y )   &    |- ( ph -> R Er X )   &    |- ( ph -> X e. _V )   =>    |- ( ph -> F C_ ( ( X /. R ) X. Y ) )
Theorem	qliftel 6862*	Elementhood in the relation F. (Contributed by Mario Carneiro, 23-Dec-2016.)
|- F = ran ( x e. X |-> <. [ x ] R , A >. )   &    |- ( ( ph /\ x e. X ) -> A e. Y )   &    |- ( ph -> R Er X )   &    |- ( ph -> X e. _V )   =>    |- ( ph -> ( [ C ] R F D <-> E. x e. X ( C R x /\ D = A ) ) )
Theorem	qliftel1 6863*	Elementhood in the relation F. (Contributed by Mario Carneiro, 23-Dec-2016.)
|- F = ran ( x e. X |-> <. [ x ] R , A >. )   &    |- ( ( ph /\ x e. X ) -> A e. Y )   &    |- ( ph -> R Er X )   &    |- ( ph -> X e. _V )   =>    |- ( ( ph /\ x e. X ) -> [ x ] R F A )
Theorem	qliftfun 6864*	The function F is the unique function defined by F ` [ x ] = A, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
|- F = ran ( x e. X |-> <. [ x ] R , A >. )   &    |- ( ( ph /\ x e. X ) -> A e. Y )   &    |- ( ph -> R Er X )   &    |- ( ph -> X e. _V )   &    |- ( x = y -> A = B )   =>    |- ( ph -> ( Fun F <-> A. x A. y ( x R y -> A = B ) ) )
Theorem	qliftfund 6865*	The function F is the unique function defined by F ` [ x ] = A, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
|- F = ran ( x e. X |-> <. [ x ] R , A >. )   &    |- ( ( ph /\ x e. X ) -> A e. Y )   &    |- ( ph -> R Er X )   &    |- ( ph -> X e. _V )   &    |- ( x = y -> A = B )   &    |- ( ( ph /\ x R y ) -> A = B )   =>    |- ( ph -> Fun F )
Theorem	qliftfuns 6866*	The function F is the unique function defined by F ` [ x ] = A, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
|- F = ran ( x e. X |-> <. [ x ] R , A >. )   &    |- ( ( ph /\ x e. X ) -> A e. Y )   &    |- ( ph -> R Er X )   &    |- ( ph -> X e. _V )   =>    |- ( ph -> ( Fun F <-> A. y A. z ( y R z -> [_ y / x ]_ A = [_ z / x ]_ A ) ) )
Theorem	qliftf 6867*	The domain and codomain of the function F. (Contributed by Mario Carneiro, 23-Dec-2016.)
|- F = ran ( x e. X |-> <. [ x ] R , A >. )   &    |- ( ( ph /\ x e. X ) -> A e. Y )   &    |- ( ph -> R Er X )   &    |- ( ph -> X e. _V )   =>    |- ( ph -> ( Fun F <-> F : ( X /. R ) --> Y ) )
Theorem	qliftval 6868*	The value of the function F. (Contributed by Mario Carneiro, 23-Dec-2016.)
|- F = ran ( x e. X |-> <. [ x ] R , A >. )   &    |- ( ( ph /\ x e. X ) -> A e. Y )   &    |- ( ph -> R Er X )   &    |- ( ph -> X e. _V )   &    |- ( x = C -> A = B )   &    |- ( ph -> Fun F )   =>    |- ( ( ph /\ C e. X ) -> ( F ` [ C ] R ) = B )
Theorem	ecoptocl 6869*	Implicit substitution of class for equivalence class of ordered pair. (Contributed by NM, 23-Jul-1995.)
|- S = ( ( B X. C ) /. R )   &    |- ( [ <. x , y >. ] R = A -> ( ph <-> ps ) )   &    |- ( ( x e. B /\ y e. C ) -> ph )   =>    |- ( A e. S -> ps )
Theorem	2ecoptocl 6870*	Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 23-Jul-1995.)
|- S = ( ( C X. D ) /. R )   &    |- ( [ <. x , y >. ] R = A -> ( ph <-> ps ) )   &    |- ( [ <. z , w >. ] R = B -> ( ps <-> ch ) )   &    |- ( ( ( x e. C /\ y e. D ) /\ ( z e. C /\ w e. D ) ) -> ph )   =>    |- ( ( A e. S /\ B e. S ) -> ch )
Theorem	3ecoptocl 6871*	Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 9-Aug-1995.)
|- S = ( ( D X. D ) /. R )   &    |- ( [ <. x , y >. ] R = A -> ( ph <-> ps ) )   &    |- ( [ <. z , w >. ] R = B -> ( ps <-> ch ) )   &    |- ( [ <. v , u >. ] R = C -> ( ch <-> th ) )   &    |- ( ( ( x e. D /\ y e. D ) /\ ( z e. D /\ w e. D ) /\ ( v e. D /\ u e. D ) ) -> ph )   =>    |- ( ( A e. S /\ B e. S /\ C e. S ) -> th )
Theorem	brecop 6872*	Binary relation on a quotient set. Lemma for real number construction. (Contributed by NM, 29-Jan-1996.)
|- .~ e. _V   &    |- .~ Er ( G X. G )   &    |- H = ( ( G X. G ) /. .~ )   &    |- .<_ = { <. x , y >. | ( ( x e. H /\ y e. H ) /\ E. z E. w E. v E. u ( ( x = [ <. z , w >. ] .~ /\ y = [ <. v , u >. ] .~ ) /\ ph ) ) }   &    |- ( ( ( ( z e. G /\ w e. G ) /\ ( A e. G /\ B e. G ) ) /\ ( ( v e. G /\ u e. G ) /\ ( C e. G /\ D e. G ) ) ) -> ( ( [ <. z , w >. ] .~ = [ <. A , B >. ] .~ /\ [ <. v , u >. ] .~ = [ <. C , D >. ] .~ ) -> ( ph <-> ps ) ) )   =>    |- ( ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) -> ( [ <. A , B >. ] .~ .<_ [ <. C , D >. ] .~ <-> ps ) )
Theorem	eroveu 6873*	Lemma for eroprf 6875. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)
|- J = ( A /. R )   &    |- K = ( B /. S )   &    |- ( ph -> T e. Z )   &    |- ( ph -> R Er U )   &    |- ( ph -> S Er V )   &    |- ( ph -> T Er W )   &    |- ( ph -> A C_ U )   &    |- ( ph -> B C_ V )   &    |- ( ph -> C C_ W )   &    |- ( ph -> .+ : ( A X. B ) --> C )   &    |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) ) -> ( ( r R s /\ t S u ) -> ( r .+ t ) T ( s .+ u ) ) )   =>    |- ( ( ph /\ ( X e. J /\ Y e. K ) ) -> E! z E. p e. A E. q e. B ( ( X = [ p ] R /\ Y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) )
Theorem	erovlem 6874*	Lemma for eroprf 6875. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)
|- J = ( A /. R )   &    |- K = ( B /. S )   &    |- ( ph -> T e. Z )   &    |- ( ph -> R Er U )   &    |- ( ph -> S Er V )   &    |- ( ph -> T Er W )   &    |- ( ph -> A C_ U )   &    |- ( ph -> B C_ V )   &    |- ( ph -> C C_ W )   &    |- ( ph -> .+ : ( A X. B ) --> C )   &    |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) ) -> ( ( r R s /\ t S u ) -> ( r .+ t ) T ( s .+ u ) ) )   &    |- .+^ = { <. <. x , y >. , z >. | E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) }   =>    |- ( ph -> .+^ = ( x e. J , y e. K |-> ( iota z E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) ) )
Theorem	eroprf 6875*	Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)
|- J = ( A /. R )   &    |- K = ( B /. S )   &    |- ( ph -> T e. Z )   &    |- ( ph -> R Er U )   &    |- ( ph -> S Er V )   &    |- ( ph -> T Er W )   &    |- ( ph -> A C_ U )   &    |- ( ph -> B C_ V )   &    |- ( ph -> C C_ W )   &    |- ( ph -> .+ : ( A X. B ) --> C )   &    |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) ) -> ( ( r R s /\ t S u ) -> ( r .+ t ) T ( s .+ u ) ) )   &    |- .+^ = { <. <. x , y >. , z >. | E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) }   &    |- ( ph -> R e. X )   &    |- ( ph -> S e. Y )   &    |- L = ( C /. T )   =>    |- ( ph -> .+^ : ( J X. K ) --> L )
Theorem	eroprf2 6876*	Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.)
|- J = ( A /. .~ )   &    |- .+^ = { <. <. x , y >. , z >. | E. p e. A E. q e. A ( ( x = [ p ] .~ /\ y = [ q ] .~ ) /\ z = [ ( p .+ q ) ] .~ ) }   &    |- ( ph -> .~ e. X )   &    |- ( ph -> .~ Er U )   &    |- ( ph -> A C_ U )   &    |- ( ph -> .+ : ( A X. A ) --> A )   &    |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. A /\ u e. A ) ) ) -> ( ( r .~ s /\ t .~ u ) -> ( r .+ t ) .~ ( s .+ u ) ) )   =>    |- ( ph -> .+^ : ( J X. J ) --> J )
Theorem	ecopoveq 6877*	This is the first of several theorems about equivalence relations of the kind used in construction of fractions and signed reals, involving operations on equivalent classes of ordered pairs. This theorem expresses the relation .~ (specified by the hypothesis) in terms of its operation F. (Contributed by NM, 16-Aug-1995.)
|- .~ = { <. x , y >. | ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z .+ u ) = ( w .+ v ) ) ) }   =>    |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> ( <. A , B >. .~ <. C , D >. <-> ( A .+ D ) = ( B .+ C ) ) )
Theorem	ecopovsym 6878*	Assuming the operation F is commutative, show that the relation .~, specified by the first hypothesis, is symmetric. (Contributed by NM, 27-Aug-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- .~ = { <. x , y >. | ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z .+ u ) = ( w .+ v ) ) ) }   &    |- ( x .+ y ) = ( y .+ x )   =>    |- ( A .~ B -> B .~ A )
Theorem	ecopovtrn 6879*	Assuming that operation F is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation .~, specified by the first hypothesis, is transitive. (Contributed by NM, 11-Feb-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- .~ = { <. x , y >. | ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z .+ u ) = ( w .+ v ) ) ) }   &    |- ( x .+ y ) = ( y .+ x )   &    |- ( ( x e. S /\ y e. S ) -> ( x .+ y ) e. S )   &    |- ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) )   &    |- ( ( x e. S /\ y e. S ) -> ( ( x .+ y ) = ( x .+ z ) -> y = z ) )   =>    |- ( ( A .~ B /\ B .~ C ) -> A .~ C )
Theorem	ecopover 6880*	Assuming that operation F is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation .~, specified by the first hypothesis, is an equivalence relation. (Contributed by NM, 16-Feb-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)
|- .~ = { <. x , y >. | ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z .+ u ) = ( w .+ v ) ) ) }   &    |- ( x .+ y ) = ( y .+ x )   &    |- ( ( x e. S /\ y e. S ) -> ( x .+ y ) e. S )   &    |- ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) )   &    |- ( ( x e. S /\ y e. S ) -> ( ( x .+ y ) = ( x .+ z ) -> y = z ) )   =>    |- .~ Er ( S X. S )
Theorem	ecopovsymg 6881*	Assuming the operation F is commutative, show that the relation .~, specified by the first hypothesis, is symmetric. (Contributed by Jim Kingdon, 1-Sep-2019.)
|- .~ = { <. x , y >. | ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z .+ u ) = ( w .+ v ) ) ) }   &    |- ( ( x e. S /\ y e. S ) -> ( x .+ y ) = ( y .+ x ) )   =>    |- ( A .~ B -> B .~ A )
Theorem	ecopovtrng 6882*	Assuming that operation F is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation .~, specified by the first hypothesis, is transitive. (Contributed by Jim Kingdon, 1-Sep-2019.)
|- .~ = { <. x , y >. | ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z .+ u ) = ( w .+ v ) ) ) }   &    |- ( ( x e. S /\ y e. S ) -> ( x .+ y ) = ( y .+ x ) )   &    |- ( ( x e. S /\ y e. S ) -> ( x .+ y ) e. S )   &    |- ( ( x e. S /\ y e. S /\ z e. S ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ( x e. S /\ y e. S /\ z e. S ) -> ( ( x .+ y ) = ( x .+ z ) -> y = z ) )   =>    |- ( ( A .~ B /\ B .~ C ) -> A .~ C )
Theorem	ecopoverg 6883*	Assuming that operation F is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation .~, specified by the first hypothesis, is an equivalence relation. (Contributed by Jim Kingdon, 1-Sep-2019.)
|- .~ = { <. x , y >. | ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z .+ u ) = ( w .+ v ) ) ) }   &    |- ( ( x e. S /\ y e. S ) -> ( x .+ y ) = ( y .+ x ) )   &    |- ( ( x e. S /\ y e. S ) -> ( x .+ y ) e. S )   &    |- ( ( x e. S /\ y e. S /\ z e. S ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ( x e. S /\ y e. S /\ z e. S ) -> ( ( x .+ y ) = ( x .+ z ) -> y = z ) )   =>    |- .~ Er ( S X. S )
Theorem	th3qlem1 6884*	Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60. The third hypothesis is the compatibility assumption. (Contributed by NM, 3-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
|- .~ Er S   &    |- ( ( ( y e. S /\ w e. S ) /\ ( z e. S /\ v e. S ) ) -> ( ( y .~ w /\ z .~ v ) -> ( y .+ z ) .~ ( w .+ v ) ) )   =>    |- ( ( A e. ( S /. .~ ) /\ B e. ( S /. .~ ) ) -> E* x E. y E. z ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ x = [ ( y .+ z ) ] .~ ) )
Theorem	th3qlem2 6885*	Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. The fourth hypothesis is the compatibility assumption. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
|- .~ e. _V   &    |- .~ Er ( S X. S )   &    |- ( ( ( ( w e. S /\ v e. S ) /\ ( u e. S /\ t e. S ) ) /\ ( ( s e. S /\ f e. S ) /\ ( g e. S /\ h e. S ) ) ) -> ( ( <. w , v >. .~ <. u , t >. /\ <. s , f >. .~ <. g , h >. ) -> ( <. w , v >. .+ <. s , f >. ) .~ ( <. u , t >. .+ <. g , h >. ) ) )   =>    |- ( ( A e. ( ( S X. S ) /. .~ ) /\ B e. ( ( S X. S ) /. .~ ) ) -> E* z E. w E. v E. u E. t ( ( A = [ <. w , v >. ] .~ /\ B = [ <. u , t >. ] .~ ) /\ z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) )
Theorem	th3qcor 6886*	Corollary of Theorem 3Q of [Enderton] p. 60. (Contributed by NM, 12-Nov-1995.) (Revised by David Abernethy, 4-Jun-2013.)
|- .~ e. _V   &    |- .~ Er ( S X. S )   &    |- ( ( ( ( w e. S /\ v e. S ) /\ ( u e. S /\ t e. S ) ) /\ ( ( s e. S /\ f e. S ) /\ ( g e. S /\ h e. S ) ) ) -> ( ( <. w , v >. .~ <. u , t >. /\ <. s , f >. .~ <. g , h >. ) -> ( <. w , v >. .+ <. s , f >. ) .~ ( <. u , t >. .+ <. g , h >. ) ) )   &    |- G = { <. <. x , y >. , z >. | ( ( x e. ( ( S X. S ) /. .~ ) /\ y e. ( ( S X. S ) /. .~ ) ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] .~ /\ y = [ <. u , t >. ] .~ ) /\ z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) ) }   =>    |- Fun G
Theorem	th3q 6887*	Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 19-Dec-2013.)
|- .~ e. _V   &    |- .~ Er ( S X. S )   &    |- ( ( ( ( w e. S /\ v e. S ) /\ ( u e. S /\ t e. S ) ) /\ ( ( s e. S /\ f e. S ) /\ ( g e. S /\ h e. S ) ) ) -> ( ( <. w , v >. .~ <. u , t >. /\ <. s , f >. .~ <. g , h >. ) -> ( <. w , v >. .+ <. s , f >. ) .~ ( <. u , t >. .+ <. g , h >. ) ) )   &    |- G = { <. <. x , y >. , z >. | ( ( x e. ( ( S X. S ) /. .~ ) /\ y e. ( ( S X. S ) /. .~ ) ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] .~ /\ y = [ <. u , t >. ] .~ ) /\ z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) ) }   =>    |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> ( [ <. A , B >. ] .~ G [ <. C , D >. ] .~ ) = [ ( <. A , B >. .+ <. C , D >. ) ] .~ )
Theorem	oviec 6888*	Express an operation on equivalence classes of ordered pairs in terms of equivalence class of operations on ordered pairs. See iset.mm for additional comments describing the hypotheses. (Unnecessary distinct variable restrictions were removed by David Abernethy, 4-Jun-2013.) (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 4-Jun-2013.)
|- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> H e. ( S X. S ) )   &    |- ( ( ( a e. S /\ b e. S ) /\ ( g e. S /\ h e. S ) ) -> K e. ( S X. S ) )   &    |- ( ( ( c e. S /\ d e. S ) /\ ( t e. S /\ s e. S ) ) -> L e. ( S X. S ) )   &    |- .~ e. _V   &    |- .~ Er ( S X. S )   &    |- .~ = { <. x , y >. | ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ph ) ) }   &    |- ( ( ( z = a /\ w = b ) /\ ( v = c /\ u = d ) ) -> ( ph <-> ps ) )   &    |- ( ( ( z = g /\ w = h ) /\ ( v = t /\ u = s ) ) -> ( ph <-> ch ) )   &    |- .+ = { <. <. x , y >. , z >. | ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = J ) ) }   &    |- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> J = K )   &    |- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> J = L )   &    |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> J = H )   &    |- .+^ = { <. <. x , y >. , z >. | ( ( x e. Q /\ y e. Q ) /\ E. a E. b E. c E. d ( ( x = [ <. a , b >. ] .~ /\ y = [ <. c , d >. ] .~ ) /\ z = [ ( <. a , b >. .+ <. c , d >. ) ] .~ ) ) }   &    |- Q = ( ( S X. S ) /. .~ )   &    |- ( ( ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) /\ ( ( g e. S /\ h e. S ) /\ ( t e. S /\ s e. S ) ) ) -> ( ( ps /\ ch ) -> K .~ L ) )   =>    |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> ( [ <. A , B >. ] .~ .+^ [ <. C , D >. ] .~ ) = [ H ] .~ )
Theorem	ecovcom 6889*	Lemma used to transfer a commutative law via an equivalence relation. Most uses will want ecovicom 6890 instead. (Contributed by NM, 29-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.)
|- C = ( ( S X. S ) /. .~ )   &    |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) = [ <. D , G >. ] .~ )   &    |- ( ( ( z e. S /\ w e. S ) /\ ( x e. S /\ y e. S ) ) -> ( [ <. z , w >. ] .~ .+ [ <. x , y >. ] .~ ) = [ <. H , J >. ] .~ )   &    |- D = H   &    |- G = J   =>    |- ( ( A e. C /\ B e. C ) -> ( A .+ B ) = ( B .+ A ) )
Theorem	ecovicom 6890*	Lemma used to transfer a commutative law via an equivalence relation. (Contributed by Jim Kingdon, 15-Sep-2019.)
|- C = ( ( S X. S ) /. .~ )   &    |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) = [ <. D , G >. ] .~ )   &    |- ( ( ( z e. S /\ w e. S ) /\ ( x e. S /\ y e. S ) ) -> ( [ <. z , w >. ] .~ .+ [ <. x , y >. ] .~ ) = [ <. H , J >. ] .~ )   &    |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> D = H )   &    |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> G = J )   =>    |- ( ( A e. C /\ B e. C ) -> ( A .+ B ) = ( B .+ A ) )
Theorem	ecovass 6891*	Lemma used to transfer an associative law via an equivalence relation. In most cases ecoviass 6892 will be more useful. (Contributed by NM, 31-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.)
|- D = ( ( S X. S ) /. .~ )   &    |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) = [ <. G , H >. ] .~ )   &    |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) = [ <. N , Q >. ] .~ )   &    |- ( ( ( G e. S /\ H e. S ) /\ ( v e. S /\ u e. S ) ) -> ( [ <. G , H >. ] .~ .+ [ <. v , u >. ] .~ ) = [ <. J , K >. ] .~ )   &    |- ( ( ( x e. S /\ y e. S ) /\ ( N e. S /\ Q e. S ) ) -> ( [ <. x , y >. ] .~ .+ [ <. N , Q >. ] .~ ) = [ <. L , M >. ] .~ )   &    |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( G e. S /\ H e. S ) )   &    |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> ( N e. S /\ Q e. S ) )   &    |- J = L   &    |- K = M   =>    |- ( ( A e. D /\ B e. D /\ C e. D ) -> ( ( A .+ B ) .+ C ) = ( A .+ ( B .+ C ) ) )
Theorem	ecoviass 6892*	Lemma used to transfer an associative law via an equivalence relation. (Contributed by Jim Kingdon, 16-Sep-2019.)
|- D = ( ( S X. S ) /. .~ )   &    |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) = [ <. G , H >. ] .~ )   &    |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) = [ <. N , Q >. ] .~ )   &    |- ( ( ( G e. S /\ H e. S ) /\ ( v e. S /\ u e. S ) ) -> ( [ <. G , H >. ] .~ .+ [ <. v , u >. ] .~ ) = [ <. J , K >. ] .~ )   &    |- ( ( ( x e. S /\ y e. S ) /\ ( N e. S /\ Q e. S ) ) -> ( [ <. x , y >. ] .~ .+ [ <. N , Q >. ] .~ ) = [ <. L , M >. ] .~ )   &    |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( G e. S /\ H e. S ) )   &    |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> ( N e. S /\ Q e. S ) )   &    |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> J = L )   &    |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> K = M )   =>    |- ( ( A e. D /\ B e. D /\ C e. D ) -> ( ( A .+ B ) .+ C ) = ( A .+ ( B .+ C ) ) )
Theorem	ecovdi 6893*	Lemma used to transfer a distributive law via an equivalence relation. Most likely ecovidi 6894 will be more helpful. (Contributed by NM, 2-Sep-1995.) (Revised by David Abernethy, 4-Jun-2013.)
|- D = ( ( S X. S ) /. .~ )   &    |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) = [ <. M , N >. ] .~ )   &    |- ( ( ( x e. S /\ y e. S ) /\ ( M e. S /\ N e. S ) ) -> ( [ <. x , y >. ] .~ .x. [ <. M , N >. ] .~ ) = [ <. H , J >. ] .~ )   &    |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( [ <. x , y >. ] .~ .x. [ <. z , w >. ] .~ ) = [ <. W , X >. ] .~ )   &    |- ( ( ( x e. S /\ y e. S ) /\ ( v e. S /\ u e. S ) ) -> ( [ <. x , y >. ] .~ .x. [ <. v , u >. ] .~ ) = [ <. Y , Z >. ] .~ )   &    |- ( ( ( W e. S /\ X e. S ) /\ ( Y e. S /\ Z e. S ) ) -> ( [ <. W , X >. ] .~ .+ [ <. Y , Z >. ] .~ ) = [ <. K , L >. ] .~ )   &    |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> ( M e. S /\ N e. S ) )   &    |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( W e. S /\ X e. S ) )   &    |- ( ( ( x e. S /\ y e. S ) /\ ( v e. S /\ u e. S ) ) -> ( Y e. S /\ Z e. S ) )   &    |- H = K   &    |- J = L   =>    |- ( ( A e. D /\ B e. D /\ C e. D ) -> ( A .x. ( B .+ C ) ) = ( ( A .x. B ) .+ ( A .x. C ) ) )
Theorem	ecovidi 6894*	Lemma used to transfer a distributive law via an equivalence relation. (Contributed by Jim Kingdon, 17-Sep-2019.)
|- D = ( ( S X. S ) /. .~ )   &    |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) = [ <. M , N >. ] .~ )   &    |- ( ( ( x e. S /\ y e. S ) /\ ( M e. S /\ N e. S ) ) -> ( [ <. x , y >. ] .~ .x. [ <. M , N >. ] .~ ) = [ <. H , J >. ] .~ )   &    |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( [ <. x , y >. ] .~ .x. [ <. z , w >. ] .~ ) = [ <. W , X >. ] .~ )   &    |- ( ( ( x e. S /\ y e. S ) /\ ( v e. S /\ u e. S ) ) -> ( [ <. x , y >. ] .~ .x. [ <. v , u >. ] .~ ) = [ <. Y , Z >. ] .~ )   &    |- ( ( ( W e. S /\ X e. S ) /\ ( Y e. S /\ Z e. S ) ) -> ( [ <. W , X >. ] .~ .+ [ <. Y , Z >. ] .~ ) = [ <. K , L >. ] .~ )   &    |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> ( M e. S /\ N e. S ) )   &    |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( W e. S /\ X e. S ) )   &    |- ( ( ( x e. S /\ y e. S ) /\ ( v e. S /\ u e. S ) ) -> ( Y e. S /\ Z e. S ) )   &    |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> H = K )   &    |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> J = L )   =>    |- ( ( A e. D /\ B e. D /\ C e. D ) -> ( A .x. ( B .+ C ) ) = ( ( A .x. B ) .+ ( A .x. C ) ) )
2.6.27  The mapping operation
Syntax	cmap 6895	Extend the definition of a class to include the mapping operation. (Read for A ^m B, "the set of all functions that map from B to A.)
class ^m
Syntax	cpm 6896	Extend the definition of a class to include the partial mapping operation. (Read for A ^pm B, "the set of all partial functions that map from B to A.)
class ^pm
Definition	df-map 6897*	Define the mapping operation or set exponentiation. The set of all functions that map from B to A is written ( A ^m B ) (see mapval 6907). Many authors write A followed by B as a superscript for this operation and rely on context to avoid confusion other exponentiation operations (e.g., Definition 10.42 of [TakeutiZaring] p. 95). Other authors show B as a prefixed superscript, which is read " A pre B " (e.g., definition of [Enderton] p. 52). Definition 8.21 of [Eisenberg] p. 125 uses the notation Map( B, A) for our ( A ^m B ). The up-arrow is used by Donald Knuth for iterated exponentiation (Science 194, 1235-1242, 1976). We adopt the first case of his notation (simple exponentiation) and subscript it with m to distinguish it from other kinds of exponentiation. (Contributed by NM, 8-Dec-2003.)
|- ^m = ( x e. _V , y e. _V |-> { f | f : y --> x } )
Definition	df-pm 6898*	Define the partial mapping operation. A partial function from B to A is a function from a subset of B to A. The set of all partial functions from B to A is written ( A ^pm B ) (see pmvalg 6906). A notation for this operation apparently does not appear in the literature. We use ^pm to distinguish it from the less general set exponentiation operation ^m (df-map 6897) . See mapsspm 6929 for its relationship to set exponentiation. (Contributed by NM, 15-Nov-2007.)
|- ^pm = ( x e. _V , y e. _V |-> { f e. ~P ( y X. x ) | Fun f } )
Theorem	mapprc 6899*	When A is a proper class, the class of all functions mapping A to B is empty. Exercise 4.41 of [Mendelson] p. 255. (Contributed by NM, 8-Dec-2003.)
|- ( -. A e. _V -> { f | f : A --> B } = (/) )
Theorem	pmex 6900*	The class of all partial functions from one set to another is a set. (Contributed by NM, 15-Nov-2007.)
|- ( ( A e. C /\ B e. D ) -> { f | ( Fun f /\ f C_ ( A X. B ) ) } e. _V )