P. 54 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 5301-5400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fnsng 5301	Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.)
|- ( ( A e. V /\ B e. W ) -> { <. A , B >. } Fn { A } )
Theorem	funsn 5302	A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 12-Aug-1994.)
|- A e. _V   &    |- B e. _V   =>    |- Fun { <. A , B >. }
Theorem	funinsn 5303	A function based on the singleton of an ordered pair. Unlike funsng 5300, this holds even if A or B is a proper class. (Contributed by Jim Kingdon, 17-Apr-2022.)
|- Fun ( { <. A , B >. } i^i ( V X. W ) )
Theorem	funprg 5304	A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.)
|- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) /\ A =/= B ) -> Fun { <. A , C >. , <. B , D >. } )
Theorem	funtpg 5305	A set of three pairs is a function if their first members are different. (Contributed by Alexander van der Vekens, 5-Dec-2017.)
|- ( ( ( X e. U /\ Y e. V /\ Z e. W ) /\ ( A e. F /\ B e. G /\ C e. H ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> Fun { <. X , A >. , <. Y , B >. , <. Z , C >. } )
Theorem	funpr 5306	A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- D e. _V   =>    |- ( A =/= B -> Fun { <. A , C >. , <. B , D >. } )
Theorem	funtp 5307	A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- D e. _V   &    |- E e. _V   &    |- F e. _V   =>    |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> Fun { <. A , D >. , <. B , E >. , <. C , F >. } )
Theorem	fnsn 5308	Functionality and domain of the singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- A e. _V   &    |- B e. _V   =>    |- { <. A , B >. } Fn { A }
Theorem	fnprg 5309	Function with a domain of two different values. (Contributed by FL, 26-Jun-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) /\ A =/= B ) -> { <. A , C >. , <. B , D >. } Fn { A , B } )
Theorem	fntpg 5310	Function with a domain of three different values. (Contributed by Alexander van der Vekens, 5-Dec-2017.)
|- ( ( ( X e. U /\ Y e. V /\ Z e. W ) /\ ( A e. F /\ B e. G /\ C e. H ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> { <. X , A >. , <. Y , B >. , <. Z , C >. } Fn { X , Y , Z } )
Theorem	fntp 5311	A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- D e. _V   &    |- E e. _V   &    |- F e. _V   =>    |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> { <. A , D >. , <. B , E >. , <. C , F >. } Fn { A , B , C } )
Theorem	fun0 5312	The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.)
|- Fun (/)
Theorem	funcnvcnv 5313	The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.)
|- ( Fun A -> Fun `' `' A )
Theorem	funcnv2 5314*	A simpler equivalence for single-rooted (see funcnv 5315). (Contributed by NM, 9-Aug-2004.)
|- ( Fun `' A <-> A. y E* x x A y )
Theorem	funcnv 5315*	The converse of a class is a function iff the class is single-rooted, which means that for any y in the range of A there is at most one x such that x A y. Definition of single-rooted in [Enderton] p. 43. See funcnv2 5314 for a simpler version. (Contributed by NM, 13-Aug-2004.)
|- ( Fun `' A <-> A. y e. ran A E* x x A y )
Theorem	funcnv3 5316*	A condition showing a class is single-rooted. (See funcnv 5315). (Contributed by NM, 26-May-2006.)
|- ( Fun `' A <-> A. y e. ran A E! x e. dom A x A y )
Theorem	funcnveq 5317*	Another way of expressing that a class is single-rooted. Counterpart to dffun2 5264. (Contributed by Jim Kingdon, 24-Dec-2018.)
|- ( Fun `' A <-> A. x A. y A. z ( ( x A y /\ z A y ) -> x = z ) )
Theorem	fun2cnv 5318*	The double converse of a class is a function iff the class is single-valued. Each side is equivalent to Definition 6.4(2) of [TakeutiZaring] p. 23, who use the notation "Un(A)" for single-valued. Note that A is not necessarily a function. (Contributed by NM, 13-Aug-2004.)
|- ( Fun `' `' A <-> A. x E* y x A y )
Theorem	svrelfun 5319	A single-valued relation is a function. (See fun2cnv 5318 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 17-Jan-2006.)
|- ( Fun A <-> ( Rel A /\ Fun `' `' A ) )
Theorem	fncnv 5320*	Single-rootedness (see funcnv 5315) of a class cut down by a cross product. (Contributed by NM, 5-Mar-2007.)
|- ( `' ( R i^i ( A X. B ) ) Fn B <-> A. y e. B E! x e. A x R y )
Theorem	fun11 5321*	Two ways of stating that A is one-to-one (but not necessarily a function). Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un2 (A)" for one-to-one (but not necessarily a function). (Contributed by NM, 17-Jan-2006.)
|- ( ( Fun `' `' A /\ Fun `' A ) <-> A. x A. y A. z A. w ( ( x A y /\ z A w ) -> ( x = z <-> y = w ) ) )
Theorem	fununi 5322*	The union of a chain (with respect to inclusion) of functions is a function. (Contributed by NM, 10-Aug-2004.)
|- ( A. f e. A ( Fun f /\ A. g e. A ( f C_ g \/ g C_ f ) ) -> Fun U. A )
Theorem	funcnvuni 5323*	The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 5315 for "single-rooted" definition.) (Contributed by NM, 11-Aug-2004.)
|- ( A. f e. A ( Fun `' f /\ A. g e. A ( f C_ g \/ g C_ f ) ) -> Fun `' U. A )
Theorem	fun11uni 5324*	The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function. (Contributed by NM, 11-Aug-2004.)
|- ( A. f e. A ( ( Fun f /\ Fun `' f ) /\ A. g e. A ( f C_ g \/ g C_ f ) ) -> ( Fun U. A /\ Fun `' U. A ) )
Theorem	funin 5325	The intersection with a function is a function. Exercise 14(a) of [Enderton] p. 53. (Contributed by NM, 19-Mar-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|- ( Fun F -> Fun ( F i^i G ) )
Theorem	funres11 5326	The restriction of a one-to-one function is one-to-one. (Contributed by NM, 25-Mar-1998.)
|- ( Fun `' F -> Fun `' ( F |` A ) )
Theorem	funcnvres 5327	The converse of a restricted function. (Contributed by NM, 27-Mar-1998.)
|- ( Fun `' F -> `' ( F |` A ) = ( `' F |` ( F " A ) ) )
Theorem	cnvresid 5328	Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.)
|- `' ( _I |` A ) = ( _I |` A )
Theorem	funcnvres2 5329	The converse of a restriction of the converse of a function equals the function restricted to the image of its converse. (Contributed by NM, 4-May-2005.)
|- ( Fun F -> `' ( `' F |` A ) = ( F |` ( `' F " A ) ) )
Theorem	funimacnv 5330	The image of the preimage of a function. (Contributed by NM, 25-May-2004.)
|- ( Fun F -> ( F " ( `' F " A ) ) = ( A i^i ran F ) )
Theorem	funimass1 5331	A kind of contraposition law that infers a subclass of an image from a preimage subclass. (Contributed by NM, 25-May-2004.)
|- ( ( Fun F /\ A C_ ran F ) -> ( ( `' F " A ) C_ B -> A C_ ( F " B ) ) )
Theorem	funimass2 5332	A kind of contraposition law that infers an image subclass from a subclass of a preimage. (Contributed by NM, 25-May-2004.)
|- ( ( Fun F /\ A C_ ( `' F " B ) ) -> ( F " A ) C_ B )
Theorem	imadiflem 5333	One direction of imadif 5334. This direction does not require Fun `' F. (Contributed by Jim Kingdon, 25-Dec-2018.)
|- ( ( F " A ) \ ( F " B ) ) C_ ( F " ( A \ B ) )
Theorem	imadif 5334	The image of a difference is the difference of images. (Contributed by NM, 24-May-1998.)
|- ( Fun `' F -> ( F " ( A \ B ) ) = ( ( F " A ) \ ( F " B ) ) )
Theorem	imainlem 5335	One direction of imain 5336. This direction does not require Fun `' F. (Contributed by Jim Kingdon, 25-Dec-2018.)
|- ( F " ( A i^i B ) ) C_ ( ( F " A ) i^i ( F " B ) )
Theorem	imain 5336	The image of an intersection is the intersection of images. (Contributed by Paul Chapman, 11-Apr-2009.)
|- ( Fun `' F -> ( F " ( A i^i B ) ) = ( ( F " A ) i^i ( F " B ) ) )
Theorem	funimaexglem 5337	Lemma for funimaexg 5338. It constitutes the interesting part of funimaexg 5338, in which B C_ dom A. (Contributed by Jim Kingdon, 27-Dec-2018.)
|- ( ( Fun A /\ B e. C /\ B C_ dom A ) -> ( A " B ) e. _V )
Theorem	funimaexg 5338	Axiom of Replacement using abbreviations. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM, 10-Sep-2006.)
|- ( ( Fun A /\ B e. C ) -> ( A " B ) e. _V )
Theorem	funimaex 5339	The image of a set under any function is also a set. Equivalent of Axiom of Replacement. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM, 17-Nov-2002.)
|- B e. _V   =>    |- ( Fun A -> ( A " B ) e. _V )
Theorem	isarep1 5340*	Part of a study of the Axiom of Replacement used by the Isabelle prover. The object PrimReplace is apparently the image of the function encoded by ph ( x , y ) i.e. the class ( { <. x , y >. | ph } " A ). If so, we can prove Isabelle's "Axiom of Replacement" conclusion without using the Axiom of Replacement, for which I (N. Megill) currently have no explanation. (Contributed by NM, 26-Oct-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
|- ( b e. ( { <. x , y >. | ph } " A ) <-> E. x e. A [ b / y ] ph )
Theorem	isarep2 5341*	Part of a study of the Axiom of Replacement used by the Isabelle prover. In Isabelle, the sethood of PrimReplace is apparently postulated implicitly by its type signature " [ i, [ i, i ] => o ] => i", which automatically asserts that it is a set without using any axioms. To prove that it is a set in Metamath, we need the hypotheses of Isabelle's "Axiom of Replacement" as well as the Axiom of Replacement in the form funimaex 5339. (Contributed by NM, 26-Oct-2006.)
|- A e. _V   &    |- A. x e. A A. y A. z ( ( ph /\ [ z / y ] ph ) -> y = z )   =>    |- E. w w = ( { <. x , y >. | ph } " A )
Theorem	fneq1 5342	Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
|- ( F = G -> ( F Fn A <-> G Fn A ) )
Theorem	fneq2 5343	Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
|- ( A = B -> ( F Fn A <-> F Fn B ) )
Theorem	fneq1d 5344	Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ph -> F = G )   =>    |- ( ph -> ( F Fn A <-> G Fn A ) )
Theorem	fneq2d 5345	Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ph -> A = B )   =>    |- ( ph -> ( F Fn A <-> F Fn B ) )
Theorem	fneq12d 5346	Equality deduction for function predicate with domain. (Contributed by NM, 26-Jun-2011.)
|- ( ph -> F = G )   &    |- ( ph -> A = B )   =>    |- ( ph -> ( F Fn A <-> G Fn B ) )
Theorem	fneq12 5347	Equality theorem for function predicate with domain. (Contributed by Thierry Arnoux, 31-Jan-2017.)
|- ( ( F = G /\ A = B ) -> ( F Fn A <-> G Fn B ) )
Theorem	fneq1i 5348	Equality inference for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
|- F = G   =>    |- ( F Fn A <-> G Fn A )
Theorem	fneq2i 5349	Equality inference for function predicate with domain. (Contributed by NM, 4-Sep-2011.)
|- A = B   =>    |- ( F Fn A <-> F Fn B )
Theorem	nffn 5350	Bound-variable hypothesis builder for a function with domain. (Contributed by NM, 30-Jan-2004.)
|- F/_ x F   &    |- F/_ x A   =>    |- F/ x F Fn A
Theorem	fnfun 5351	A function with domain is a function. (Contributed by NM, 1-Aug-1994.)
|- ( F Fn A -> Fun F )
Theorem	fnrel 5352	A function with domain is a relation. (Contributed by NM, 1-Aug-1994.)
|- ( F Fn A -> Rel F )
Theorem	fndm 5353	The domain of a function. (Contributed by NM, 2-Aug-1994.)
|- ( F Fn A -> dom F = A )
Theorem	funfni 5354	Inference to convert a function and domain antecedent. (Contributed by NM, 22-Apr-2004.)
|- ( ( Fun F /\ B e. dom F ) -> ph )   =>    |- ( ( F Fn A /\ B e. A ) -> ph )
Theorem	fndmu 5355	A function has a unique domain. (Contributed by NM, 11-Aug-1994.)
|- ( ( F Fn A /\ F Fn B ) -> A = B )
Theorem	fnbr 5356	The first argument of binary relation on a function belongs to the function's domain. (Contributed by NM, 7-May-2004.)
|- ( ( F Fn A /\ B F C ) -> B e. A )
Theorem	fnop 5357	The first argument of an ordered pair in a function belongs to the function's domain. (Contributed by NM, 8-Aug-1994.)
|- ( ( F Fn A /\ <. B , C >. e. F ) -> B e. A )
Theorem	fneu 5358*	There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|- ( ( F Fn A /\ B e. A ) -> E! y B F y )
Theorem	fneu2 5359*	There is exactly one value of a function. (Contributed by NM, 7-Nov-1995.)
|- ( ( F Fn A /\ B e. A ) -> E! y <. B , y >. e. F )
Theorem	fnun 5360	The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
|- ( ( ( F Fn A /\ G Fn B ) /\ ( A i^i B ) = (/) ) -> ( F u. G ) Fn ( A u. B ) )
Theorem	fnunsn 5361	Extension of a function with a new ordered pair. (Contributed by NM, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
|- ( ph -> X e. _V )   &    |- ( ph -> Y e. _V )   &    |- ( ph -> F Fn D )   &    |- G = ( F u. { <. X , Y >. } )   &    |- E = ( D u. { X } )   &    |- ( ph -> -. X e. D )   =>    |- ( ph -> G Fn E )
Theorem	fnco 5362	Composition of two functions. (Contributed by NM, 22-May-2006.)
|- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> ( F o. G ) Fn B )
Theorem	fnresdm 5363	A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.)
|- ( F Fn A -> ( F |` A ) = F )
Theorem	fnresdisj 5364	A function restricted to a class disjoint with its domain is empty. (Contributed by NM, 23-Sep-2004.)
|- ( F Fn A -> ( ( A i^i B ) = (/) <-> ( F |` B ) = (/) ) )
Theorem	2elresin 5365	Membership in two functions restricted by each other's domain. (Contributed by NM, 8-Aug-1994.)
|- ( ( F Fn A /\ G Fn B ) -> ( ( <. x , y >. e. F /\ <. x , z >. e. G ) <-> ( <. x , y >. e. ( F |` ( A i^i B ) ) /\ <. x , z >. e. ( G |` ( A i^i B ) ) ) ) )
Theorem	fnssresb 5366	Restriction of a function with a subclass of its domain. (Contributed by NM, 10-Oct-2007.)
|- ( F Fn A -> ( ( F |` B ) Fn B <-> B C_ A ) )
Theorem	fnssres 5367	Restriction of a function with a subclass of its domain. (Contributed by NM, 2-Aug-1994.)
|- ( ( F Fn A /\ B C_ A ) -> ( F |` B ) Fn B )
Theorem	fnresin1 5368	Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
|- ( F Fn A -> ( F |` ( A i^i B ) ) Fn ( A i^i B ) )
Theorem	fnresin2 5369	Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
|- ( F Fn A -> ( F |` ( B i^i A ) ) Fn ( B i^i A ) )
Theorem	fnres 5370*	An equivalence for functionality of a restriction. Compare dffun8 5282. (Contributed by Mario Carneiro, 20-May-2015.)
|- ( ( F |` A ) Fn A <-> A. x e. A E! y x F y )
Theorem	fnresi 5371	Functionality and domain of restricted identity. (Contributed by NM, 27-Aug-2004.)
|- ( _I |` A ) Fn A
Theorem	fnima 5372	The image of a function's domain is its range. (Contributed by NM, 4-Nov-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|- ( F Fn A -> ( F " A ) = ran F )
Theorem	fn0 5373	A function with empty domain is empty. (Contributed by NM, 15-Apr-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|- ( F Fn (/) <-> F = (/) )
Theorem	fnimadisj 5374	A class that is disjoint with the domain of a function has an empty image under the function. (Contributed by FL, 24-Jan-2007.)
|- ( ( F Fn A /\ ( A i^i C ) = (/) ) -> ( F " C ) = (/) )
Theorem	fnimaeq0 5375	Images under a function never map nonempty sets to empty sets. (Contributed by Stefan O'Rear, 21-Jan-2015.)
|- ( ( F Fn A /\ B C_ A ) -> ( ( F " B ) = (/) <-> B = (/) ) )
Theorem	dfmpt3 5376	Alternate definition for the maps-to notation df-mpt 4092. (Contributed by Mario Carneiro, 30-Dec-2016.)
|- ( x e. A |-> B ) = U_ x e. A ( { x } X. { B } )
Theorem	fnopabg 5377*	Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
|- F = { <. x , y >. | ( x e. A /\ ph ) }   =>    |- ( A. x e. A E! y ph <-> F Fn A )
Theorem	fnopab 5378*	Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 5-Mar-1996.)
|- ( x e. A -> E! y ph )   &    |- F = { <. x , y >. | ( x e. A /\ ph ) }   =>    |- F Fn A
Theorem	mptfng 5379*	The maps-to notation defines a function with domain. (Contributed by Scott Fenton, 21-Mar-2011.)
|- F = ( x e. A |-> B )   =>    |- ( A. x e. A B e. _V <-> F Fn A )
Theorem	fnmpt 5380*	The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.)
|- F = ( x e. A |-> B )   =>    |- ( A. x e. A B e. V -> F Fn A )
Theorem	mpt0 5381	A mapping operation with empty domain. (Contributed by Mario Carneiro, 28-Dec-2014.)
|- ( x e. (/) |-> A ) = (/)
Theorem	fnmpti 5382*	Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- B e. _V   &    |- F = ( x e. A |-> B )   =>    |- F Fn A
Theorem	dmmpti 5383*	Domain of an ordered-pair class abstraction that specifies a function. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- B e. _V   &    |- F = ( x e. A |-> B )   =>    |- dom F = A
Theorem	dmmptd 5384*	The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- A = ( x e. B |-> C )   &    |- ( ( ph /\ x e. B ) -> C e. V )   =>    |- ( ph -> dom A = B )
Theorem	mptun 5385	Union of mappings which are mutually compatible. (Contributed by Mario Carneiro, 31-Aug-2015.)
|- ( x e. ( A u. B ) |-> C ) = ( ( x e. A |-> C ) u. ( x e. B |-> C ) )
Theorem	feq1 5386	Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
|- ( F = G -> ( F : A --> B <-> G : A --> B ) )
Theorem	feq2 5387	Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
|- ( A = B -> ( F : A --> C <-> F : B --> C ) )
Theorem	feq3 5388	Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
|- ( A = B -> ( F : C --> A <-> F : C --> B ) )
Theorem	feq23 5389	Equality theorem for functions. (Contributed by FL, 14-Jul-2007.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|- ( ( A = C /\ B = D ) -> ( F : A --> B <-> F : C --> D ) )
Theorem	feq1d 5390	Equality deduction for functions. (Contributed by NM, 19-Feb-2008.)
|- ( ph -> F = G )   =>    |- ( ph -> ( F : A --> B <-> G : A --> B ) )
Theorem	feq2d 5391	Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ph -> A = B )   =>    |- ( ph -> ( F : A --> C <-> F : B --> C ) )
Theorem	feq3d 5392	Equality deduction for functions. (Contributed by AV, 1-Jan-2020.)
|- ( ph -> A = B )   =>    |- ( ph -> ( F : X --> A <-> F : X --> B ) )
Theorem	feq12d 5393	Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ph -> F = G )   &    |- ( ph -> A = B )   =>    |- ( ph -> ( F : A --> C <-> G : B --> C ) )
Theorem	feq123d 5394	Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ph -> F = G )   &    |- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( F : A --> C <-> G : B --> D ) )
Theorem	feq123 5395	Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
|- ( ( F = G /\ A = C /\ B = D ) -> ( F : A --> B <-> G : C --> D ) )
Theorem	feq1i 5396	Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
|- F = G   =>    |- ( F : A --> B <-> G : A --> B )
Theorem	feq2i 5397	Equality inference for functions. (Contributed by NM, 5-Sep-2011.)
|- A = B   =>    |- ( F : A --> C <-> F : B --> C )
Theorem	feq23i 5398	Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
|- A = C   &    |- B = D   =>    |- ( F : A --> B <-> F : C --> D )
Theorem	feq23d 5399	Equality deduction for functions. (Contributed by NM, 8-Jun-2013.)
|- ( ph -> A = C )   &    |- ( ph -> B = D )   =>    |- ( ph -> ( F : A --> B <-> F : C --> D ) )
Theorem	nff 5400	Bound-variable hypothesis builder for a mapping. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- F/_ x F   &    |- F/_ x A   &    |- F/_ x B   =>    |- F/ x F : A --> B