P. 67 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6601-6700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ectocld 6601*	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 6602*	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 6603*	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 6604	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 6605	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 6606	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 6607*	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 6608*	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 6609	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 6610	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 6611	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 6612	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 6613*	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 6614*	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 6615*	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 6616*	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 6617*	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 6618*	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 6619*	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 6620*	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 6621*	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 6622*	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 6623*	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 6624*	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 6625*	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 6626*	Lemma for eroprf 6628. (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 6627*	Lemma for eroprf 6628. (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 6628*	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 6629*	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 6630*	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 6631*	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 6632*	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 6633*	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 6634*	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 6635*	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 6636*	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 6637*	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 6638*	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 6639*	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 6640*	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 6641*	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 6642*	Lemma used to transfer a commutative law via an equivalence relation. Most uses will want ecovicom 6643 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 6643*	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 6644*	Lemma used to transfer an associative law via an equivalence relation. In most cases ecoviass 6645 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 6645*	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 6646*	Lemma used to transfer a distributive law via an equivalence relation. Most likely ecovidi 6647 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 6647*	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 ) ) )
2.6.26  The mapping operation
Syntax	cmap 6648	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 6649	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 6650*	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 6660). 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 6651*	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 6659). 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 6650) . See mapsspm 6682 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 6652*	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 6653*	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 )
Theorem	mapex 6654*	The class of all functions mapping one set to another is a set. Remark after Definition 10.24 of [Kunen] p. 31. (Contributed by Raph Levien, 4-Dec-2003.)
|- ( ( A e. C /\ B e. D ) -> { f | f : A --> B } e. _V )
Theorem	fnmap 6655	Set exponentiation has a universal domain. (Contributed by NM, 8-Dec-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)
|- ^m Fn ( _V X. _V )
Theorem	fnpm 6656	Partial function exponentiation has a universal domain. (Contributed by Mario Carneiro, 14-Nov-2013.)
|- ^pm Fn ( _V X. _V )
Theorem	reldmmap 6657	Set exponentiation is a well-behaved binary operator. (Contributed by Stefan O'Rear, 27-Feb-2015.)
|- Rel dom ^m
Theorem	mapvalg 6658*	The value of set exponentiation. ( A ^m B ) is the set of all functions that map from B to A. Definition 10.24 of [Kunen] p. 24. (Contributed by NM, 8-Dec-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)
|- ( ( A e. C /\ B e. D ) -> ( A ^m B ) = { f | f : B --> A } )
Theorem	pmvalg 6659*	The value of the partial mapping operation. ( A ^pm B ) is the set of all partial functions that map from B to A. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 8-Sep-2013.)
|- ( ( A e. C /\ B e. D ) -> ( A ^pm B ) = { f e. ~P ( B X. A ) | Fun f } )
Theorem	mapval 6660*	The value of set exponentiation (inference version). ( A ^m B ) is the set of all functions that map from B to A. Definition 10.24 of [Kunen] p. 24. (Contributed by NM, 8-Dec-2003.)
|- A e. _V   &    |- B e. _V   =>    |- ( A ^m B ) = { f | f : B --> A }
Theorem	elmapg 6661	Membership relation for set exponentiation. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 15-Nov-2014.)
|- ( ( A e. V /\ B e. W ) -> ( C e. ( A ^m B ) <-> C : B --> A ) )
Theorem	elmapd 6662	Deduction form of elmapg 6661. (Contributed by BJ, 11-Apr-2020.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   =>    |- ( ph -> ( C e. ( A ^m B ) <-> C : B --> A ) )
Theorem	mapdm0 6663	The empty set is the only map with empty domain. (Contributed by Glauco Siliprandi, 11-Oct-2020.) (Proof shortened by Thierry Arnoux, 3-Dec-2021.)
|- ( B e. V -> ( B ^m (/) ) = { (/) } )
Theorem	elpmg 6664	The predicate "is a partial function". (Contributed by Mario Carneiro, 14-Nov-2013.)
|- ( ( A e. V /\ B e. W ) -> ( C e. ( A ^pm B ) <-> ( Fun C /\ C C_ ( B X. A ) ) ) )
Theorem	elpm2g 6665	The predicate "is a partial function". (Contributed by NM, 31-Dec-2013.)
|- ( ( A e. V /\ B e. W ) -> ( F e. ( A ^pm B ) <-> ( F : dom F --> A /\ dom F C_ B ) ) )
Theorem	elpm2r 6666	Sufficient condition for being a partial function. (Contributed by NM, 31-Dec-2013.)
|- ( ( ( A e. V /\ B e. W ) /\ ( F : C --> A /\ C C_ B ) ) -> F e. ( A ^pm B ) )
Theorem	elpmi 6667	A partial function is a function. (Contributed by Mario Carneiro, 15-Sep-2015.)
|- ( F e. ( A ^pm B ) -> ( F : dom F --> A /\ dom F C_ B ) )
Theorem	pmfun 6668	A partial function is a function. (Contributed by Mario Carneiro, 30-Jan-2014.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( F e. ( A ^pm B ) -> Fun F )
Theorem	elmapex 6669	Eliminate antecedent for mapping theorems: domain can be taken to be a set. (Contributed by Stefan O'Rear, 8-Oct-2014.)
|- ( A e. ( B ^m C ) -> ( B e. _V /\ C e. _V ) )
Theorem	elmapi 6670	A mapping is a function, forward direction only with superfluous antecedent removed. (Contributed by Stefan O'Rear, 10-Oct-2014.)
|- ( A e. ( B ^m C ) -> A : C --> B )
Theorem	elmapfn 6671	A mapping is a function with the appropriate domain. (Contributed by AV, 6-Apr-2019.)
|- ( A e. ( B ^m C ) -> A Fn C )
Theorem	elmapfun 6672	A mapping is always a function. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
|- ( A e. ( B ^m C ) -> Fun A )
Theorem	elmapssres 6673	A restricted mapping is a mapping. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Mario Carneiro, 5-May-2015.)
|- ( ( A e. ( B ^m C ) /\ D C_ C ) -> ( A |` D ) e. ( B ^m D ) )
Theorem	fpmg 6674	A total function is a partial function. (Contributed by Mario Carneiro, 31-Dec-2013.)
|- ( ( A e. V /\ B e. W /\ F : A --> B ) -> F e. ( B ^pm A ) )
Theorem	pmss12g 6675	Subset relation for the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)
|- ( ( ( A C_ C /\ B C_ D ) /\ ( C e. V /\ D e. W ) ) -> ( A ^pm B ) C_ ( C ^pm D ) )
Theorem	pmresg 6676	Elementhood of a restricted function in the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)
|- ( ( B e. V /\ F e. ( A ^pm C ) ) -> ( F |` B ) e. ( A ^pm B ) )
Theorem	elmap 6677	Membership relation for set exponentiation. (Contributed by NM, 8-Dec-2003.)
|- A e. _V   &    |- B e. _V   =>    |- ( F e. ( A ^m B ) <-> F : B --> A )
Theorem	mapval2 6678*	Alternate expression for the value of set exponentiation. (Contributed by NM, 3-Nov-2007.)
|- A e. _V   &    |- B e. _V   =>    |- ( A ^m B ) = ( ~P ( B X. A ) i^i { f | f Fn B } )
Theorem	elpm 6679	The predicate "is a partial function". (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 14-Nov-2013.)
|- A e. _V   &    |- B e. _V   =>    |- ( F e. ( A ^pm B ) <-> ( Fun F /\ F C_ ( B X. A ) ) )
Theorem	elpm2 6680	The predicate "is a partial function". (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 31-Dec-2013.)
|- A e. _V   &    |- B e. _V   =>    |- ( F e. ( A ^pm B ) <-> ( F : dom F --> A /\ dom F C_ B ) )
Theorem	fpm 6681	A total function is a partial function. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 31-Dec-2013.)
|- A e. _V   &    |- B e. _V   =>    |- ( F : A --> B -> F e. ( B ^pm A ) )
Theorem	mapsspm 6682	Set exponentiation is a subset of partial maps. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 27-Feb-2016.)
|- ( A ^m B ) C_ ( A ^pm B )
Theorem	pmsspw 6683	Partial maps are a subset of the power set of the Cartesian product of its arguments. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- ( A ^pm B ) C_ ~P ( B X. A )
Theorem	mapsspw 6684	Set exponentiation is a subset of the power set of the Cartesian product of its arguments. (Contributed by NM, 8-Dec-2006.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( A ^m B ) C_ ~P ( B X. A )
Theorem	fvmptmap 6685*	Special case of fvmpt 5594 for operator theorems. (Contributed by NM, 27-Nov-2007.)
|- C e. _V   &    |- D e. _V   &    |- R e. _V   &    |- ( x = A -> B = C )   &    |- F = ( x e. ( R ^m D ) |-> B )   =>    |- ( A : D --> R -> ( F ` A ) = C )
Theorem	map0e 6686	Set exponentiation with an empty exponent (ordinal number 0) is ordinal number 1. Exercise 4.42(a) of [Mendelson] p. 255. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 30-Apr-2015.)
|- ( A e. V -> ( A ^m (/) ) = 1o )
Theorem	map0b 6687	Set exponentiation with an empty base is the empty set, provided the exponent is nonempty. Theorem 96 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( A =/= (/) -> ( (/) ^m A ) = (/) )
Theorem	map0g 6688	Set exponentiation is empty iff the base is empty and the exponent is not empty. Theorem 97 of [Suppes] p. 89. (Contributed by Mario Carneiro, 30-Apr-2015.)
|- ( ( A e. V /\ B e. W ) -> ( ( A ^m B ) = (/) <-> ( A = (/) /\ B =/= (/) ) ) )
Theorem	map0 6689	Set exponentiation is empty iff the base is empty and the exponent is not empty. Theorem 97 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.)
|- A e. _V   &    |- B e. _V   =>    |- ( ( A ^m B ) = (/) <-> ( A = (/) /\ B =/= (/) ) )
Theorem	mapsn 6690*	The value of set exponentiation with a singleton exponent. Theorem 98 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.)
|- A e. _V   &    |- B e. _V   =>    |- ( A ^m { B } ) = { f | E. y e. A f = { <. B , y >. } }
Theorem	mapss 6691	Subset inheritance for set exponentiation. Theorem 99 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( ( B e. V /\ A C_ B ) -> ( A ^m C ) C_ ( B ^m C ) )
Theorem	fdiagfn 6692*	Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)
|- F = ( x e. B |-> ( I X. { x } ) )   =>    |- ( ( B e. V /\ I e. W ) -> F : B --> ( B ^m I ) )
Theorem	fvdiagfn 6693*	Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)
|- F = ( x e. B |-> ( I X. { x } ) )   =>    |- ( ( I e. W /\ X e. B ) -> ( F ` X ) = ( I X. { X } ) )
Theorem	mapsnconst 6694	Every singleton map is a constant function. (Contributed by Stefan O'Rear, 25-Mar-2015.)
|- S = { X }   &    |- B e. _V   &    |- X e. _V   =>    |- ( F e. ( B ^m S ) -> F = ( S X. { ( F ` X ) } ) )
Theorem	mapsncnv 6695*	Expression for the inverse of the canonical map between a set and its set of singleton functions. (Contributed by Stefan O'Rear, 21-Mar-2015.)
|- S = { X }   &    |- B e. _V   &    |- X e. _V   &    |- F = ( x e. ( B ^m S ) |-> ( x ` X ) )   =>    |- `' F = ( y e. B |-> ( S X. { y } ) )
Theorem	mapsnf1o2 6696*	Explicit bijection between a set and its singleton functions. (Contributed by Stefan O'Rear, 21-Mar-2015.)
|- S = { X }   &    |- B e. _V   &    |- X e. _V   &    |- F = ( x e. ( B ^m S ) |-> ( x ` X ) )   =>    |- F : ( B ^m S ) -1-1-onto-> B
Theorem	mapsnf1o3 6697*	Explicit bijection in the reverse of mapsnf1o2 6696. (Contributed by Stefan O'Rear, 24-Mar-2015.)
|- S = { X }   &    |- B e. _V   &    |- X e. _V   &    |- F = ( y e. B |-> ( S X. { y } ) )   =>    |- F : B -1-1-onto-> ( B ^m S )
2.6.27  Infinite Cartesian products
Syntax	cixp 6698	Extend class notation to include infinite Cartesian products.
class X_ x e. A B
Definition	df-ixp 6699*	Definition of infinite Cartesian product of [Enderton] p. 54. Enderton uses a bold "X" with x e. A written underneath or as a subscript, as does Stoll p. 47. Some books use a capital pi, but we will reserve that notation for products of numbers. Usually B represents a class expression containing x free and thus can be thought of as B ( x ). Normally, x is not free in A, although this is not a requirement of the definition. (Contributed by NM, 28-Sep-2006.)
|- X_ x e. A B = { f | ( f Fn { x | x e. A } /\ A. x e. A ( f ` x ) e. B ) }
Theorem	dfixp 6700*	Eliminate the expression { x | x e. A } in df-ixp 6699, under the assumption that A and x are disjoint. This way, we can say that x is bound in X_ x e. A B even if it appears free in A. (Contributed by Mario Carneiro, 12-Aug-2016.)
|- X_ x e. A B = { f | ( f Fn A /\ A. x e. A ( f ` x ) e. B ) }