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	f1ofn 5501	A one-to-one onto mapping is function on its domain. (Contributed by NM, 12-Dec-2003.)
|- ( F : A -1-1-onto-> B -> F Fn A )
Theorem	f1ofun 5502	A one-to-one onto mapping is a function. (Contributed by NM, 12-Dec-2003.)
|- ( F : A -1-1-onto-> B -> Fun F )
Theorem	f1orel 5503	A one-to-one onto mapping is a relation. (Contributed by NM, 13-Dec-2003.)
|- ( F : A -1-1-onto-> B -> Rel F )
Theorem	f1odm 5504	The domain of a one-to-one onto mapping. (Contributed by NM, 8-Mar-2014.)
|- ( F : A -1-1-onto-> B -> dom F = A )
Theorem	dff1o2 5505	Alternate definition of one-to-one onto function. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|- ( F : A -1-1-onto-> B <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )
Theorem	dff1o3 5506	Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|- ( F : A -1-1-onto-> B <-> ( F : A -onto-> B /\ Fun `' F ) )
Theorem	f1ofo 5507	A one-to-one onto function is an onto function. (Contributed by NM, 28-Apr-2004.)
|- ( F : A -1-1-onto-> B -> F : A -onto-> B )
Theorem	dff1o4 5508	Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) )
Theorem	dff1o5 5509	Alternate definition of one-to-one onto function. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ ran F = B ) )
Theorem	f1orn 5510	A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.)
|- ( F : A -1-1-onto-> ran F <-> ( F Fn A /\ Fun `' F ) )
Theorem	f1f1orn 5511	A one-to-one function maps one-to-one onto its range. (Contributed by NM, 4-Sep-2004.)
|- ( F : A -1-1-> B -> F : A -1-1-onto-> ran F )
Theorem	f1oabexg 5512*	The class of all 1-1-onto functions mapping one set to another is a set. (Contributed by Paul Chapman, 25-Feb-2008.)
|- F = { f | ( f : A -1-1-onto-> B /\ ph ) }   =>    |- ( ( A e. C /\ B e. D ) -> F e. _V )
Theorem	f1ocnv 5513	The converse of a one-to-one onto function is also one-to-one onto. (Contributed by NM, 11-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|- ( F : A -1-1-onto-> B -> `' F : B -1-1-onto-> A )
Theorem	f1ocnvb 5514	A relation is a one-to-one onto function iff its converse is a one-to-one onto function with domain and codomain/range interchanged. (Contributed by NM, 8-Dec-2003.)
|- ( Rel F -> ( F : A -1-1-onto-> B <-> `' F : B -1-1-onto-> A ) )
Theorem	f1ores 5515	The restriction of a one-to-one function maps one-to-one onto the image. (Contributed by NM, 25-Mar-1998.)
|- ( ( F : A -1-1-> B /\ C C_ A ) -> ( F |` C ) : C -1-1-onto-> ( F " C ) )
Theorem	f1orescnv 5516	The converse of a one-to-one-onto restricted function. (Contributed by Paul Chapman, 21-Apr-2008.)
|- ( ( Fun `' F /\ ( F |` R ) : R -1-1-onto-> P ) -> ( `' F |` P ) : P -1-1-onto-> R )
Theorem	f1imacnv 5517	Preimage of an image. (Contributed by NM, 30-Sep-2004.)
|- ( ( F : A -1-1-> B /\ C C_ A ) -> ( `' F " ( F " C ) ) = C )
Theorem	foimacnv 5518	A reverse version of f1imacnv 5517. (Contributed by Jeff Hankins, 16-Jul-2009.)
|- ( ( F : A -onto-> B /\ C C_ B ) -> ( F " ( `' F " C ) ) = C )
Theorem	foun 5519	The union of two onto functions with disjoint domains is an onto function. (Contributed by Mario Carneiro, 22-Jun-2016.)
|- ( ( ( F : A -onto-> B /\ G : C -onto-> D ) /\ ( A i^i C ) = (/) ) -> ( F u. G ) : ( A u. C ) -onto-> ( B u. D ) )
Theorem	f1oun 5520	The union of two one-to-one onto functions with disjoint domains and ranges. (Contributed by NM, 26-Mar-1998.)
|- ( ( ( F : A -1-1-onto-> B /\ G : C -1-1-onto-> D ) /\ ( ( A i^i C ) = (/) /\ ( B i^i D ) = (/) ) ) -> ( F u. G ) : ( A u. C ) -1-1-onto-> ( B u. D ) )
Theorem	fun11iun 5521*	The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function. (Contributed by Mario Carneiro, 20-May-2013.) (Revised by Mario Carneiro, 24-Jun-2015.)
|- ( x = y -> B = C )   &    |- B e. _V   =>    |- ( A. x e. A ( B : D -1-1-> S /\ A. y e. A ( B C_ C \/ C C_ B ) ) -> U_ x e. A B : U_ x e. A D -1-1-> S )
Theorem	resdif 5522	The restriction of a one-to-one onto function to a difference maps onto the difference of the images. (Contributed by Paul Chapman, 11-Apr-2009.)
|- ( ( Fun `' F /\ ( F |` A ) : A -onto-> C /\ ( F |` B ) : B -onto-> D ) -> ( F |` ( A \ B ) ) : ( A \ B ) -1-1-onto-> ( C \ D ) )
Theorem	f1oco 5523	Composition of one-to-one onto functions. (Contributed by NM, 19-Mar-1998.)
|- ( ( F : B -1-1-onto-> C /\ G : A -1-1-onto-> B ) -> ( F o. G ) : A -1-1-onto-> C )
Theorem	f1cnv 5524	The converse of an injective function is bijective. (Contributed by FL, 11-Nov-2011.)
|- ( F : A -1-1-> B -> `' F : ran F -1-1-onto-> A )
Theorem	funcocnv2 5525	Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( Fun F -> ( F o. `' F ) = ( _I |` ran F ) )
Theorem	fococnv2 5526	The composition of an onto function and its converse. (Contributed by Stefan O'Rear, 12-Feb-2015.)
|- ( F : A -onto-> B -> ( F o. `' F ) = ( _I |` B ) )
Theorem	f1ococnv2 5527	The composition of a one-to-one onto function and its converse equals the identity relation restricted to the function's range. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Stefan O'Rear, 12-Feb-2015.)
|- ( F : A -1-1-onto-> B -> ( F o. `' F ) = ( _I |` B ) )
Theorem	f1cocnv2 5528	Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
|- ( F : A -1-1-> B -> ( F o. `' F ) = ( _I |` ran F ) )
Theorem	f1ococnv1 5529	The composition of a one-to-one onto function's converse and itself equals the identity relation restricted to the function's domain. (Contributed by NM, 13-Dec-2003.)
|- ( F : A -1-1-onto-> B -> ( `' F o. F ) = ( _I |` A ) )
Theorem	f1cocnv1 5530	Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
|- ( F : A -1-1-> B -> ( `' F o. F ) = ( _I |` A ) )
Theorem	funcoeqres 5531	Express a constraint on a composition as a constraint on the composand. (Contributed by Stefan O'Rear, 7-Mar-2015.)
|- ( ( Fun G /\ ( F o. G ) = H ) -> ( F |` ran G ) = ( H o. `' G ) )
Theorem	ffoss 5532*	Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145. (Contributed by NM, 10-May-1998.)
|- F e. _V   =>    |- ( F : A --> B <-> E. x ( F : A -onto-> x /\ x C_ B ) )
Theorem	f11o 5533*	Relationship between one-to-one and one-to-one onto function. (Contributed by NM, 4-Apr-1998.)
|- F e. _V   =>    |- ( F : A -1-1-> B <-> E. x ( F : A -1-1-onto-> x /\ x C_ B ) )
Theorem	f10 5534	The empty set maps one-to-one into any class. (Contributed by NM, 7-Apr-1998.)
|- (/) : (/) -1-1-> A
Theorem	f1o00 5535	One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.)
|- ( F : (/) -1-1-onto-> A <-> ( F = (/) /\ A = (/) ) )
Theorem	fo00 5536	Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.)
|- ( F : (/) -onto-> A <-> ( F = (/) /\ A = (/) ) )
Theorem	f1o0 5537	One-to-one onto mapping of the empty set. (Contributed by NM, 10-Sep-2004.)
|- (/) : (/) -1-1-onto-> (/)
Theorem	f1oi 5538	A restriction of the identity relation is a one-to-one onto function. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|- ( _I |` A ) : A -1-1-onto-> A
Theorem	f1ovi 5539	The identity relation is a one-to-one onto function on the universe. (Contributed by NM, 16-May-2004.)
|- _I : _V -1-1-onto-> _V
Theorem	f1osn 5540	A singleton of an ordered pair is one-to-one onto function. (Contributed by NM, 18-May-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|- A e. _V   &    |- B e. _V   =>    |- { <. A , B >. } : { A } -1-1-onto-> { B }
Theorem	f1osng 5541	A singleton of an ordered pair is one-to-one onto function. (Contributed by Mario Carneiro, 12-Jan-2013.)
|- ( ( A e. V /\ B e. W ) -> { <. A , B >. } : { A } -1-1-onto-> { B } )
Theorem	f1sng 5542	A singleton of an ordered pair is a one-to-one function. (Contributed by AV, 17-Apr-2021.)
|- ( ( A e. V /\ B e. W ) -> { <. A , B >. } : { A } -1-1-> W )
Theorem	fsnd 5543	A singleton of an ordered pair is a function. (Contributed by AV, 17-Apr-2021.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   =>    |- ( ph -> { <. A , B >. } : { A } --> W )
Theorem	f1oprg 5544	An unordered pair of ordered pairs with different elements is a one-to-one onto function. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
|- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) -> ( ( A =/= C /\ B =/= D ) -> { <. A , B >. , <. C , D >. } : { A , C } -1-1-onto-> { B , D } ) )
Theorem	tz6.12-2 5545*	Function value when F is not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
|- ( -. E! x A F x -> ( F ` A ) = (/) )
Theorem	fveu 5546*	The value of a function at a unique point. (Contributed by Scott Fenton, 6-Oct-2017.)
|- ( E! x A F x -> ( F ` A ) = U. { x | A F x } )
Theorem	brprcneu 5547*	If A is a proper class and F is any class, then there is no unique set which is related to A through the binary relation F. (Contributed by Scott Fenton, 7-Oct-2017.)
|- ( -. A e. _V -> -. E! x A F x )
Theorem	fvprc 5548	A function's value at a proper class is the empty set. (Contributed by NM, 20-May-1998.)
|- ( -. A e. _V -> ( F ` A ) = (/) )
Theorem	fv2 5549*	Alternate definition of function value. Definition 10.11 of [Quine] p. 68. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- ( F ` A ) = U. { x | A. y ( A F y <-> y = x ) }
Theorem	dffv3g 5550*	A definition of function value in terms of iota. (Contributed by Jim Kingdon, 29-Dec-2018.)
|- ( A e. V -> ( F ` A ) = ( iota x x e. ( F " { A } ) ) )
Theorem	dffv4g 5551*	The previous definition of function value, from before the iota operator was introduced. Although based on the idea embodied by Definition 10.2 of [Quine] p. 65 (see args 5034), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.)
|- ( A e. V -> ( F ` A ) = U. { x | ( F " { A } ) = { x } } )
Theorem	elfv 5552*	Membership in a function value. (Contributed by NM, 30-Apr-2004.)
|- ( A e. ( F ` B ) <-> E. x ( A e. x /\ A. y ( B F y <-> y = x ) ) )
Theorem	fveq1 5553	Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
|- ( F = G -> ( F ` A ) = ( G ` A ) )
Theorem	fveq2 5554	Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
|- ( A = B -> ( F ` A ) = ( F ` B ) )
Theorem	fveq1i 5555	Equality inference for function value. (Contributed by NM, 2-Sep-2003.)
|- F = G   =>    |- ( F ` A ) = ( G ` A )
Theorem	fveq1d 5556	Equality deduction for function value. (Contributed by NM, 2-Sep-2003.)
|- ( ph -> F = G )   =>    |- ( ph -> ( F ` A ) = ( G ` A ) )
Theorem	fveq2i 5557	Equality inference for function value. (Contributed by NM, 28-Jul-1999.)
|- A = B   =>    |- ( F ` A ) = ( F ` B )
Theorem	fveq2d 5558	Equality deduction for function value. (Contributed by NM, 29-May-1999.)
|- ( ph -> A = B )   =>    |- ( ph -> ( F ` A ) = ( F ` B ) )
Theorem	2fveq3 5559	Equality theorem for nested function values. (Contributed by AV, 14-Aug-2022.)
|- ( A = B -> ( F ` ( G ` A ) ) = ( F ` ( G ` B ) ) )
Theorem	fveq12i 5560	Equality deduction for function value. (Contributed by FL, 27-Jun-2014.)
|- F = G   &    |- A = B   =>    |- ( F ` A ) = ( G ` B )
Theorem	fveq12d 5561	Equality deduction for function value. (Contributed by FL, 22-Dec-2008.)
|- ( ph -> F = G )   &    |- ( ph -> A = B )   =>    |- ( ph -> ( F ` A ) = ( G ` B ) )
Theorem	fveqeq2d 5562	Equality deduction for function value. (Contributed by BJ, 30-Aug-2022.)
|- ( ph -> A = B )   =>    |- ( ph -> ( ( F ` A ) = C <-> ( F ` B ) = C ) )
Theorem	fveqeq2 5563	Equality deduction for function value. (Contributed by BJ, 31-Aug-2022.)
|- ( A = B -> ( ( F ` A ) = C <-> ( F ` B ) = C ) )
Theorem	nffv 5564	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 5565*	Bound-variable hypothesis builder for mapping, special case. (Contributed by Mario Carneiro, 25-Dec-2016.)
|- F/_ x ( ( x e. A |-> B ) ` C )
Theorem	nffvd 5566	Deduction version of bound-variable hypothesis builder nffv 5564. (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 5567*	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 5568*	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 5569	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 5570	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 5571	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 5572	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 5573	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 5574	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 5575	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 5576	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 5577*	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 5578	The value of a restricted function. (Contributed by NM, 2-Aug-1994.)
|- ( A e. B -> ( ( F |` B ) ` A ) = ( F ` A ) )
Theorem	fvresd 5579	The value of a restricted function, deduction version of fvres 5578. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
|- ( ph -> A e. B )   =>    |- ( ph -> ( ( F |` B ) ` A ) = ( F ` A ) )
Theorem	funssfv 5580	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 5581*	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 5582*	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 5583*	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 5584*	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 5585	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 5586	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	elfvm 5587*	If a function value has a member, the function is inhabited. (Contributed by Jim Kingdon, 14-Jun-2025.)
|- ( A e. ( F ` B ) -> E. j j e. F )
Theorem	nfvres 5588	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 5589	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 5590	Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.)
|- ( (/) ` A ) = (/)
Theorem	fv2prc 5591	A function value of a function value at a proper class is the empty set. (Contributed by AV, 8-Apr-2021.)
|- ( -. A e. _V -> ( ( F ` A ) ` B ) = (/) )
Theorem	csbfv12g 5592	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 5593*	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 5594*	Substitution for a function value. (Contributed by NM, 1-Jan-2006.)
|- ( A e. C -> [_ A / x ]_ ( F ` x ) = ( F ` A ) )
Theorem	funbrfv 5595	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 5596	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 5597	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 5598	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 5599	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 5600	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 ) )