P. 56 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 5501-5600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	2fveq3 5501	Equality theorem for nested function values. (Contributed by AV, 14-Aug-2022.)
|- ( A = B -> ( F ` ( G ` A ) ) = ( F ` ( G ` B ) ) )
Theorem	fveq12i 5502	Equality deduction for function value. (Contributed by FL, 27-Jun-2014.)
|- F = G   &    |- A = B   =>    |- ( F ` A ) = ( G ` B )
Theorem	fveq12d 5503	Equality deduction for function value. (Contributed by FL, 22-Dec-2008.)
|- ( ph -> F = G )   &    |- ( ph -> A = B )   =>    |- ( ph -> ( F ` A ) = ( G ` B ) )
Theorem	fveqeq2d 5504	Equality deduction for function value. (Contributed by BJ, 30-Aug-2022.)
|- ( ph -> A = B )   =>    |- ( ph -> ( ( F ` A ) = C <-> ( F ` B ) = C ) )
Theorem	fveqeq2 5505	Equality deduction for function value. (Contributed by BJ, 31-Aug-2022.)
|- ( A = B -> ( ( F ` A ) = C <-> ( F ` B ) = C ) )
Theorem	nffv 5506	Bound-variable hypothesis builder for function value. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- F/_ x F   &    |- F/_ x A   =>    |- F/_ x ( F ` A )
Theorem	nffvmpt1 5507*	Bound-variable hypothesis builder for mapping, special case. (Contributed by Mario Carneiro, 25-Dec-2016.)
|- F/_ x ( ( x e. A |-> B ) ` C )
Theorem	nffvd 5508	Deduction version of bound-variable hypothesis builder nffv 5506. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- ( ph -> F/_ x F )   &    |- ( ph -> F/_ x A )   =>    |- ( ph -> F/_ x ( F ` A ) )
Theorem	funfveu 5509*	A function has one value given an argument in its domain. (Contributed by Jim Kingdon, 29-Dec-2018.)
|- ( ( Fun F /\ A e. dom F ) -> E! y A F y )
Theorem	fvss 5510*	The value of a function is a subset of B if every element that could be a candidate for the value is a subset of B. (Contributed by Mario Carneiro, 24-May-2019.)
|- ( A. x ( A F x -> x C_ B ) -> ( F ` A ) C_ B )
Theorem	fvssunirng 5511	The result of a function value is always a subset of the union of the range, if the input is a set. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 24-May-2019.)
|- ( A e. _V -> ( F ` A ) C_ U. ran F )
Theorem	relfvssunirn 5512	The result of a function value is always a subset of the union of the range, even if it is invalid and thus empty. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 24-May-2019.)
|- ( Rel F -> ( F ` A ) C_ U. ran F )
Theorem	funfvex 5513	The value of a function exists. A special case of Corollary 6.13 of [TakeutiZaring] p. 27. (Contributed by Jim Kingdon, 29-Dec-2018.)
|- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. _V )
Theorem	relrnfvex 5514	If a function has a set range, then the function value exists unconditional on the domain. (Contributed by Mario Carneiro, 24-May-2019.)
|- ( ( Rel F /\ ran F e. _V ) -> ( F ` A ) e. _V )
Theorem	fvexg 5515	Evaluating a set function at a set exists. (Contributed by Mario Carneiro and Jim Kingdon, 28-May-2019.)
|- ( ( F e. V /\ A e. W ) -> ( F ` A ) e. _V )
Theorem	fvex 5516	Evaluating a set function at a set exists. (Contributed by Mario Carneiro and Jim Kingdon, 28-May-2019.)
|- F e. V   &    |- A e. W   =>    |- ( F ` A ) e. _V
Theorem	sefvex 5517	If a function is set-like, then the function value exists if the input does. (Contributed by Mario Carneiro, 24-May-2019.)
|- ( ( `' F Se _V /\ A e. _V ) -> ( F ` A ) e. _V )
Theorem	fvifdc 5518	Move a conditional outside of a function. (Contributed by Jim Kingdon, 1-Jan-2022.)
|- (DECID ph -> ( F ` if ( ph , A , B ) ) = if ( ph , ( F ` A ) , ( F ` B ) ) )
Theorem	fv3 5519*	Alternate definition of the value of a function. Definition 6.11 of [TakeutiZaring] p. 26. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- ( F ` A ) = { x | ( E. y ( x e. y /\ A F y ) /\ E! y A F y ) }
Theorem	fvres 5520	The value of a restricted function. (Contributed by NM, 2-Aug-1994.)
|- ( A e. B -> ( ( F |` B ) ` A ) = ( F ` A ) )
Theorem	fvresd 5521	The value of a restricted function, deduction version of fvres 5520. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
|- ( ph -> A e. B )   =>    |- ( ph -> ( ( F |` B ) ` A ) = ( F ` A ) )
Theorem	funssfv 5522	The value of a member of the domain of a subclass of a function. (Contributed by NM, 15-Aug-1994.)
|- ( ( Fun F /\ G C_ F /\ A e. dom G ) -> ( F ` A ) = ( G ` A ) )
Theorem	tz6.12-1 5523*	Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
|- ( ( A F y /\ E! y A F y ) -> ( F ` A ) = y )
Theorem	tz6.12 5524*	Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 10-Jul-1994.)
|- ( ( <. A , y >. e. F /\ E! y <. A , y >. e. F ) -> ( F ` A ) = y )
Theorem	tz6.12f 5525*	Function value, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 30-Aug-1999.)
|- F/_ y F   =>    |- ( ( <. A , y >. e. F /\ E! y <. A , y >. e. F ) -> ( F ` A ) = y )
Theorem	tz6.12c 5526*	Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
|- ( E! y A F y -> ( ( F ` A ) = y <-> A F y ) )
Theorem	ndmfvg 5527	The value of a class outside its domain is the empty set. (Contributed by Jim Kingdon, 15-Jan-2019.)
|- ( ( A e. _V /\ -. A e. dom F ) -> ( F ` A ) = (/) )
Theorem	relelfvdm 5528	If a function value has a member, the argument belongs to the domain. (Contributed by Jim Kingdon, 22-Jan-2019.)
|- ( ( Rel F /\ A e. ( F ` B ) ) -> B e. dom F )
Theorem	nfvres 5529	The value of a non-member of a restriction is the empty set. (Contributed by NM, 13-Nov-1995.)
|- ( -. A e. B -> ( ( F |` B ) ` A ) = (/) )
Theorem	nfunsn 5530	If the restriction of a class to a singleton is not a function, its value is the empty set. (Contributed by NM, 8-Aug-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|- ( -. Fun ( F |` { A } ) -> ( F ` A ) = (/) )
Theorem	0fv 5531	Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.)
|- ( (/) ` A ) = (/)
Theorem	csbfv12g 5532	Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.)
|- ( A e. C -> [_ A / x ]_ ( F ` B ) = ( [_ A / x ]_ F ` [_ A / x ]_ B ) )
Theorem	csbfv2g 5533*	Move class substitution in and out of a function value. (Contributed by NM, 10-Nov-2005.)
|- ( A e. C -> [_ A / x ]_ ( F ` B ) = ( F ` [_ A / x ]_ B ) )
Theorem	csbfvg 5534*	Substitution for a function value. (Contributed by NM, 1-Jan-2006.)
|- ( A e. C -> [_ A / x ]_ ( F ` x ) = ( F ` A ) )
Theorem	funbrfv 5535	The second argument of a binary relation on a function is the function's value. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( Fun F -> ( A F B -> ( F ` A ) = B ) )
Theorem	funopfv 5536	The second element in an ordered pair member of a function is the function's value. (Contributed by NM, 19-Jul-1996.)
|- ( Fun F -> ( <. A , B >. e. F -> ( F ` A ) = B ) )
Theorem	fnbrfvb 5537	Equivalence of function value and binary relation. (Contributed by NM, 19-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( ( F Fn A /\ B e. A ) -> ( ( F ` B ) = C <-> B F C ) )
Theorem	fnopfvb 5538	Equivalence of function value and ordered pair membership. (Contributed by NM, 7-Nov-1995.)
|- ( ( F Fn A /\ B e. A ) -> ( ( F ` B ) = C <-> <. B , C >. e. F ) )
Theorem	funbrfvb 5539	Equivalence of function value and binary relation. (Contributed by NM, 26-Mar-2006.)
|- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) = B <-> A F B ) )
Theorem	funopfvb 5540	Equivalence of function value and ordered pair membership. Theorem 4.3(ii) of [Monk1] p. 42. (Contributed by NM, 26-Jan-1997.)
|- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) = B <-> <. A , B >. e. F ) )
Theorem	funbrfv2b 5541	Function value in terms of a binary relation. (Contributed by Mario Carneiro, 19-Mar-2014.)
|- ( Fun F -> ( A F B <-> ( A e. dom F /\ ( F ` A ) = B ) ) )
Theorem	dffn5im 5542*	Representation of a function in terms of its values. The converse holds given the law of the excluded middle; as it is we have most of the converse via funmpt 5236 and dmmptss 5107. (Contributed by Jim Kingdon, 31-Dec-2018.)
|- ( F Fn A -> F = ( x e. A |-> ( F ` x ) ) )
Theorem	fnrnfv 5543*	The range of a function expressed as a collection of the function's values. (Contributed by NM, 20-Oct-2005.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
|- ( F Fn A -> ran F = { y | E. x e. A y = ( F ` x ) } )
Theorem	fvelrnb 5544*	A member of a function's range is a value of the function. (Contributed by NM, 31-Oct-1995.)
|- ( F Fn A -> ( B e. ran F <-> E. x e. A ( F ` x ) = B ) )
Theorem	dfimafn 5545*	Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.)
|- ( ( Fun F /\ A C_ dom F ) -> ( F " A ) = { y | E. x e. A ( F ` x ) = y } )
Theorem	dfimafn2 5546*	Alternate definition of the image of a function as an indexed union of singletons of function values. (Contributed by Raph Levien, 20-Nov-2006.)
|- ( ( Fun F /\ A C_ dom F ) -> ( F " A ) = U_ x e. A { ( F ` x ) } )
Theorem	funimass4 5547*	Membership relation for the values of a function whose image is a subclass. (Contributed by Raph Levien, 20-Nov-2006.)
|- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A. x e. A ( F ` x ) e. B ) )
Theorem	fvelima 5548*	Function value in an image. Part of Theorem 4.4(iii) of [Monk1] p. 42. (Contributed by NM, 29-Apr-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|- ( ( Fun F /\ A e. ( F " B ) ) -> E. x e. B ( F ` x ) = A )
Theorem	feqmptd 5549*	Deduction form of dffn5im 5542. (Contributed by Mario Carneiro, 8-Jan-2015.)
|- ( ph -> F : A --> B )   =>    |- ( ph -> F = ( x e. A |-> ( F ` x ) ) )
Theorem	feqresmpt 5550*	Express a restricted function as a mapping. (Contributed by Mario Carneiro, 18-May-2016.)
|- ( ph -> F : A --> B )   &    |- ( ph -> C C_ A )   =>    |- ( ph -> ( F |` C ) = ( x e. C |-> ( F ` x ) ) )
Theorem	dffn5imf 5551*	Representation of a function in terms of its values. (Contributed by Jim Kingdon, 31-Dec-2018.)
|- F/_ x F   =>    |- ( F Fn A -> F = ( x e. A |-> ( F ` x ) ) )
Theorem	fvelimab 5552*	Function value in an image. (Contributed by NM, 20-Jan-2007.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by David Abernethy, 17-Dec-2011.)
|- ( ( F Fn A /\ B C_ A ) -> ( C e. ( F " B ) <-> E. x e. B ( F ` x ) = C ) )
Theorem	fvi 5553	The value of the identity function. (Contributed by NM, 1-May-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( A e. V -> ( _I ` A ) = A )
Theorem	fniinfv 5554*	The indexed intersection of a function's values is the intersection of its range. (Contributed by NM, 20-Oct-2005.)
|- ( F Fn A -> |^|_ x e. A ( F ` x ) = |^| ran F )
Theorem	fnsnfv 5555	Singleton of function value. (Contributed by NM, 22-May-1998.)
|- ( ( F Fn A /\ B e. A ) -> { ( F ` B ) } = ( F " { B } ) )
Theorem	fnimapr 5556	The image of a pair under a function. (Contributed by Jeff Madsen, 6-Jan-2011.)
|- ( ( F Fn A /\ B e. A /\ C e. A ) -> ( F " { B , C } ) = { ( F ` B ) , ( F ` C ) } )
Theorem	ssimaex 5557*	The existence of a subimage. (Contributed by NM, 8-Apr-2007.)
|- A e. _V   =>    |- ( ( Fun F /\ B C_ ( F " A ) ) -> E. x ( x C_ A /\ B = ( F " x ) ) )
Theorem	ssimaexg 5558*	The existence of a subimage. (Contributed by FL, 15-Apr-2007.)
|- ( ( A e. C /\ Fun F /\ B C_ ( F " A ) ) -> E. x ( x C_ A /\ B = ( F " x ) ) )
Theorem	funfvdm 5559	A simplified expression for the value of a function when we know it's a function. (Contributed by Jim Kingdon, 1-Jan-2019.)
|- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) = U. ( F " { A } ) )
Theorem	funfvdm2 5560*	The value of a function. Definition of function value in [Enderton] p. 43. (Contributed by Jim Kingdon, 1-Jan-2019.)
|- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) = U. { y | A F y } )
Theorem	funfvdm2f 5561	The value of a function. Version of funfvdm2 5560 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 1-Jan-2019.)
|- F/_ y A   &    |- F/_ y F   =>    |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) = U. { y | A F y } )
Theorem	fvun1 5562	The value of a union when the argument is in the first domain. (Contributed by Scott Fenton, 29-Jun-2013.)
|- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. A ) ) -> ( ( F u. G ) ` X ) = ( F ` X ) )
Theorem	fvun2 5563	The value of a union when the argument is in the second domain. (Contributed by Scott Fenton, 29-Jun-2013.)
|- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. B ) ) -> ( ( F u. G ) ` X ) = ( G ` X ) )
Theorem	dmfco 5564	Domains of a function composition. (Contributed by NM, 27-Jan-1997.)
|- ( ( Fun G /\ A e. dom G ) -> ( A e. dom ( F o. G ) <-> ( G ` A ) e. dom F ) )
Theorem	fvco2 5565	Value of a function composition. Similar to second part of Theorem 3H of [Enderton] p. 47. (Contributed by NM, 9-Oct-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear, 16-Oct-2014.)
|- ( ( G Fn A /\ X e. A ) -> ( ( F o. G ) ` X ) = ( F ` ( G ` X ) ) )
Theorem	fvco 5566	Value of a function composition. Similar to Exercise 5 of [TakeutiZaring] p. 28. (Contributed by NM, 22-Apr-2006.) (Proof shortened by Mario Carneiro, 26-Dec-2014.)
|- ( ( Fun G /\ A e. dom G ) -> ( ( F o. G ) ` A ) = ( F ` ( G ` A ) ) )
Theorem	fvco3 5567	Value of a function composition. (Contributed by NM, 3-Jan-2004.) (Revised by Mario Carneiro, 26-Dec-2014.)
|- ( ( G : A --> B /\ C e. A ) -> ( ( F o. G ) ` C ) = ( F ` ( G ` C ) ) )
Theorem	fvco4 5568	Value of a composition. (Contributed by BJ, 7-Jul-2022.)
|- ( ( ( K : A --> X /\ ( H o. K ) = F ) /\ ( u e. A /\ x = ( K ` u ) ) ) -> ( H ` x ) = ( F ` u ) )
Theorem	fvopab3g 5569*	Value of a function given by ordered-pair class abstraction. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ( x e. C -> E! y ph )   &    |- F = { <. x , y >. | ( x e. C /\ ph ) }   =>    |- ( ( A e. C /\ B e. D ) -> ( ( F ` A ) = B <-> ch ) )
Theorem	fvopab3ig 5570*	Value of a function given by ordered-pair class abstraction. (Contributed by NM, 23-Oct-1999.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ( x e. C -> E* y ph )   &    |- F = { <. x , y >. | ( x e. C /\ ph ) }   =>    |- ( ( A e. C /\ B e. D ) -> ( ch -> ( F ` A ) = B ) )
Theorem	fvmptss2 5571*	A mapping always evaluates to a subset of the substituted expression in the mapping, even if this is a proper class, or we are out of the domain. (Contributed by Mario Carneiro, 13-Feb-2015.) (Revised by Mario Carneiro, 3-Jul-2019.)
|- ( x = D -> B = C )   &    |- F = ( x e. A |-> B )   =>    |- ( F ` D ) C_ C
Theorem	fvmptg 5572*	Value of a function given in maps-to notation. (Contributed by NM, 2-Oct-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- ( x = A -> B = C )   &    |- F = ( x e. D |-> B )   =>    |- ( ( A e. D /\ C e. R ) -> ( F ` A ) = C )
Theorem	fvmpt 5573*	Value of a function given in maps-to notation. (Contributed by NM, 17-Aug-2011.)
|- ( x = A -> B = C )   &    |- F = ( x e. D |-> B )   &    |- C e. _V   =>    |- ( A e. D -> ( F ` A ) = C )
Theorem	fvmpts 5574*	Value of a function given in maps-to notation, using explicit class substitution. (Contributed by Scott Fenton, 17-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- F = ( x e. C |-> B )   =>    |- ( ( A e. C /\ [_ A / x ]_ B e. V ) -> ( F ` A ) = [_ A / x ]_ B )
Theorem	fvmpt3 5575*	Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Stefan O'Rear, 30-Jan-2015.)
|- ( x = A -> B = C )   &    |- F = ( x e. D |-> B )   &    |- ( x e. D -> B e. V )   =>    |- ( A e. D -> ( F ` A ) = C )
Theorem	fvmpt3i 5576*	Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Mario Carneiro, 11-Sep-2015.)
|- ( x = A -> B = C )   &    |- F = ( x e. D |-> B )   &    |- B e. _V   =>    |- ( A e. D -> ( F ` A ) = C )
Theorem	fvmptd 5577*	Deduction version of fvmpt 5573. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- ( ph -> F = ( x e. D |-> B ) )   &    |- ( ( ph /\ x = A ) -> B = C )   &    |- ( ph -> A e. D )   &    |- ( ph -> C e. V )   =>    |- ( ph -> ( F ` A ) = C )
Theorem	mptrcl 5578*	Reverse closure for a mapping: If the function value of a mapping has a member, the argument belongs to the base class of the mapping. (Contributed by AV, 4-Apr-2020.) (Revised by Jim Kingdon, 27-Mar-2023.)
|- F = ( x e. A |-> B )   =>    |- ( I e. ( F ` X ) -> X e. A )
Theorem	fvmpt2 5579*	Value of a function given by the maps-to notation. (Contributed by FL, 21-Jun-2010.)
|- F = ( x e. A |-> B )   =>    |- ( ( x e. A /\ B e. C ) -> ( F ` x ) = B )
Theorem	fvmptssdm 5580*	If all the values of the mapping are subsets of a class C, then so is any evaluation of the mapping at a value in the domain of the mapping. (Contributed by Jim Kingdon, 3-Jan-2018.)
|- F = ( x e. A |-> B )   =>    |- ( ( D e. dom F /\ A. x e. A B C_ C ) -> ( F ` D ) C_ C )
Theorem	mptfvex 5581*	Sufficient condition for a maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
|- F = ( x e. A |-> B )   =>    |- ( ( A. x B e. V /\ C e. W ) -> ( F ` C ) e. _V )
Theorem	fvmpt2d 5582*	Deduction version of fvmpt2 5579. (Contributed by Thierry Arnoux, 8-Dec-2016.)
|- ( ph -> F = ( x e. A |-> B ) )   &    |- ( ( ph /\ x e. A ) -> B e. V )   =>    |- ( ( ph /\ x e. A ) -> ( F ` x ) = B )
Theorem	fvmptdf 5583*	Alternate deduction version of fvmpt 5573, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
|- ( ph -> A e. D )   &    |- ( ( ph /\ x = A ) -> B e. V )   &    |- ( ( ph /\ x = A ) -> ( ( F ` A ) = B -> ps ) )   &    |- F/_ x F   &    |- F/ x ps   =>    |- ( ph -> ( F = ( x e. D |-> B ) -> ps ) )
Theorem	fvmptdv 5584*	Alternate deduction version of fvmpt 5573, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
|- ( ph -> A e. D )   &    |- ( ( ph /\ x = A ) -> B e. V )   &    |- ( ( ph /\ x = A ) -> ( ( F ` A ) = B -> ps ) )   =>    |- ( ph -> ( F = ( x e. D |-> B ) -> ps ) )
Theorem	fvmptdv2 5585*	Alternate deduction version of fvmpt 5573, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
|- ( ph -> A e. D )   &    |- ( ( ph /\ x = A ) -> B e. V )   &    |- ( ( ph /\ x = A ) -> B = C )   =>    |- ( ph -> ( F = ( x e. D |-> B ) -> ( F ` A ) = C ) )
Theorem	mpteqb 5586*	Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 5593. (Contributed by Mario Carneiro, 14-Nov-2014.)
|- ( A. x e. A B e. V -> ( ( x e. A |-> B ) = ( x e. A |-> C ) <-> A. x e. A B = C ) )
Theorem	fvmptt 5587*	Closed theorem form of fvmpt 5573. (Contributed by Scott Fenton, 21-Feb-2013.) (Revised by Mario Carneiro, 11-Sep-2015.)
|- ( ( A. x ( x = A -> B = C ) /\ F = ( x e. D |-> B ) /\ ( A e. D /\ C e. V ) ) -> ( F ` A ) = C )
Theorem	fvmptf 5588*	Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5572 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- F/_ x A   &    |- F/_ x C   &    |- ( x = A -> B = C )   &    |- F = ( x e. D |-> B )   =>    |- ( ( A e. D /\ C e. V ) -> ( F ` A ) = C )
Theorem	fvmptd3 5589*	Deduction version of fvmpt 5573. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
|- F = ( x e. D |-> B )   &    |- ( x = A -> B = C )   &    |- ( ph -> A e. D )   &    |- ( ph -> C e. V )   =>    |- ( ph -> ( F ` A ) = C )
Theorem	elfvmptrab1 5590*	Implications for the value of a function defined by the maps-to notation with a class abstraction as a result having an element. Here, the base set of the class abstraction depends on the argument of the function. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
|- F = ( x e. V |-> { y e. [_ x / m ]_ M | ph } )   &    |- ( X e. V -> [_ X / m ]_ M e. _V )   =>    |- ( Y e. ( F ` X ) -> ( X e. V /\ Y e. [_ X / m ]_ M ) )
Theorem	elfvmptrab 5591*	Implications for the value of a function defined by the maps-to notation with a class abstraction as a result having an element. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
|- F = ( x e. V |-> { y e. M | ph } )   &    |- ( X e. V -> M e. _V )   =>    |- ( Y e. ( F ` X ) -> ( X e. V /\ Y e. M ) )
Theorem	fvopab6 5592*	Value of a function given by ordered-pair class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
|- F = { <. x , y >. | ( ph /\ y = B ) }   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( x = A -> B = C )   =>    |- ( ( A e. D /\ C e. R /\ ps ) -> ( F ` A ) = C )
Theorem	eqfnfv 5593*	Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
|- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> A. x e. A ( F ` x ) = ( G ` x ) ) )
Theorem	eqfnfv2 5594*	Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28. (Contributed by NM, 3-Aug-1994.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- ( ( F Fn A /\ G Fn B ) -> ( F = G <-> ( A = B /\ A. x e. A ( F ` x ) = ( G ` x ) ) ) )
Theorem	eqfnfv3 5595*	Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( F Fn A /\ G Fn B ) -> ( F = G <-> ( B C_ A /\ A. x e. A ( x e. B /\ ( F ` x ) = ( G ` x ) ) ) ) )
Theorem	eqfnfvd 5596*	Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- ( ph -> F Fn A )   &    |- ( ph -> G Fn A )   &    |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( G ` x ) )   =>    |- ( ph -> F = G )
Theorem	eqfnfv2f 5597*	Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). This version of eqfnfv 5593 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.)
|- F/_ x F   &    |- F/_ x G   =>    |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> A. x e. A ( F ` x ) = ( G ` x ) ) )
Theorem	eqfunfv 5598*	Equality of functions is determined by their values. (Contributed by Scott Fenton, 19-Jun-2011.)
|- ( ( Fun F /\ Fun G ) -> ( F = G <-> ( dom F = dom G /\ A. x e. dom F ( F ` x ) = ( G ` x ) ) ) )
Theorem	fvreseq 5599*	Equality of restricted functions is determined by their values. (Contributed by NM, 3-Aug-1994.)
|- ( ( ( F Fn A /\ G Fn A ) /\ B C_ A ) -> ( ( F |` B ) = ( G |` B ) <-> A. x e. B ( F ` x ) = ( G ` x ) ) )
Theorem	fnmptfvd 5600*	A function with a given domain is a mapping defined by its function values. (Contributed by AV, 1-Mar-2019.)
|- ( ph -> M Fn A )   &    |- ( i = a -> D = C )   &    |- ( ( ph /\ i e. A ) -> D e. U )   &    |- ( ( ph /\ a e. A ) -> C e. V )   =>    |- ( ph -> ( M = ( a e. A |-> C ) <-> A. i e. A ( M ` i ) = D ) )