P. 52 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 5101-5200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	funssres 5101	The restriction of a function to the domain of a subclass equals the subclass. (Contributed by NM, 15-Aug-1994.)
|- ( ( Fun F /\ G C_ F ) -> ( F |` dom G ) = G )
Theorem	fun2ssres 5102	Equality of restrictions of a function and a subclass. (Contributed by NM, 16-Aug-1994.)
|- ( ( Fun F /\ G C_ F /\ A C_ dom G ) -> ( F |` A ) = ( G |` A ) )
Theorem	funun 5103	The union of functions with disjoint domains is a function. Theorem 4.6 of [Monk1] p. 43. (Contributed by NM, 12-Aug-1994.)
|- ( ( ( Fun F /\ Fun G ) /\ ( dom F i^i dom G ) = (/) ) -> Fun ( F u. G ) )
Theorem	funcnvsn 5104	The converse singleton of an ordered pair is a function. This is equivalent to funsn 5107 via cnvsn 4957, but stating it this way allows us to skip the sethood assumptions on A and B. (Contributed by NM, 30-Apr-2015.)
|- Fun `' { <. A , B >. }
Theorem	funsng 5105	A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 28-Jun-2011.)
|- ( ( A e. V /\ B e. W ) -> Fun { <. A , B >. } )
Theorem	fnsng 5106	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 5107	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 5108	A function based on the singleton of an ordered pair. Unlike funsng 5105, 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 5109	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 5110	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 5111	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 5112	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 5113	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 5114	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 5115	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 5116	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 5117	The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.)
|- Fun (/)
Theorem	funcnvcnv 5118	The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.)
|- ( Fun A -> Fun `' `' A )
Theorem	funcnv2 5119*	A simpler equivalence for single-rooted (see funcnv 5120). (Contributed by NM, 9-Aug-2004.)
|- ( Fun `' A <-> A. y E* x x A y )
Theorem	funcnv 5120*	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 5119 for a simpler version. (Contributed by NM, 13-Aug-2004.)
|- ( Fun `' A <-> A. y e. ran A E* x x A y )
Theorem	funcnv3 5121*	A condition showing a class is single-rooted. (See funcnv 5120). (Contributed by NM, 26-May-2006.)
|- ( Fun `' A <-> A. y e. ran A E! x e. dom A x A y )
Theorem	funcnveq 5122*	Another way of expressing that a class is single-rooted. Counterpart to dffun2 5069. (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 5123*	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 5124	A single-valued relation is a function. (See fun2cnv 5123 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 5125*	Single-rootedness (see funcnv 5120) 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 5126*	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 5127*	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 5128*	The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 5120 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 5129*	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 5130	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 5131	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 5132	The converse of a restricted function. (Contributed by NM, 27-Mar-1998.)
|- ( Fun `' F -> `' ( F |` A ) = ( `' F |` ( F " A ) ) )
Theorem	cnvresid 5133	Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.)
|- `' ( _I |` A ) = ( _I |` A )
Theorem	funcnvres2 5134	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 5135	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 5136	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 5137	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 5138	One direction of imadif 5139. This direction does not require Fun `' F. (Contributed by Jim Kingdon, 25-Dec-2018.)
|- ( ( F " A ) \ ( F " B ) ) C_ ( F " ( A \ B ) )
Theorem	imadif 5139	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 5140	One direction of imain 5141. 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 5141	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 5142	Lemma for funimaexg 5143. It constitutes the interesting part of funimaexg 5143, 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 5143	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 5144	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 5145*	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 5146*	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 5144. (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 5147	Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
|- ( F = G -> ( F Fn A <-> G Fn A ) )
Theorem	fneq2 5148	Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
|- ( A = B -> ( F Fn A <-> F Fn B ) )
Theorem	fneq1d 5149	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 5150	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 5151	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 5152	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 5153	Equality inference for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
|- F = G   =>    |- ( F Fn A <-> G Fn A )
Theorem	fneq2i 5154	Equality inference for function predicate with domain. (Contributed by NM, 4-Sep-2011.)
|- A = B   =>    |- ( F Fn A <-> F Fn B )
Theorem	nffn 5155	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 5156	A function with domain is a function. (Contributed by NM, 1-Aug-1994.)
|- ( F Fn A -> Fun F )
Theorem	fnrel 5157	A function with domain is a relation. (Contributed by NM, 1-Aug-1994.)
|- ( F Fn A -> Rel F )
Theorem	fndm 5158	The domain of a function. (Contributed by NM, 2-Aug-1994.)
|- ( F Fn A -> dom F = A )
Theorem	funfni 5159	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 5160	A function has a unique domain. (Contributed by NM, 11-Aug-1994.)
|- ( ( F Fn A /\ F Fn B ) -> A = B )
Theorem	fnbr 5161	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 5162	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 5163*	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 5164*	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 5165	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 5166	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 5167	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 5168	A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.)
|- ( F Fn A -> ( F |` A ) = F )
Theorem	fnresdisj 5169	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 5170	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 5171	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 5172	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 5173	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 5174	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 5175*	An equivalence for functionality of a restriction. Compare dffun8 5087. (Contributed by Mario Carneiro, 20-May-2015.)
|- ( ( F |` A ) Fn A <-> A. x e. A E! y x F y )
Theorem	fnresi 5176	Functionality and domain of restricted identity. (Contributed by NM, 27-Aug-2004.)
|- ( _I |` A ) Fn A
Theorem	fnima 5177	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 5178	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 5179	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 5180	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 5181	Alternate definition for the maps-to notation df-mpt 3931. (Contributed by Mario Carneiro, 30-Dec-2016.)
|- ( x e. A |-> B ) = U_ x e. A ( { x } X. { B } )
Theorem	fnopabg 5182*	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 5183*	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 5184*	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 5185*	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 5186	A mapping operation with empty domain. (Contributed by Mario Carneiro, 28-Dec-2014.)
|- ( x e. (/) |-> A ) = (/)
Theorem	fnmpti 5187*	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 5188*	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 5189*	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 5190	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 5191	Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
|- ( F = G -> ( F : A --> B <-> G : A --> B ) )
Theorem	feq2 5192	Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
|- ( A = B -> ( F : A --> C <-> F : B --> C ) )
Theorem	feq3 5193	Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
|- ( A = B -> ( F : C --> A <-> F : C --> B ) )
Theorem	feq23 5194	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 5195	Equality deduction for functions. (Contributed by NM, 19-Feb-2008.)
|- ( ph -> F = G )   =>    |- ( ph -> ( F : A --> B <-> G : A --> B ) )
Theorem	feq2d 5196	Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ph -> A = B )   =>    |- ( ph -> ( F : A --> C <-> F : B --> C ) )
Theorem	feq3d 5197	Equality deduction for functions. (Contributed by AV, 1-Jan-2020.)
|- ( ph -> A = B )   =>    |- ( ph -> ( F : X --> A <-> F : X --> B ) )
Theorem	feq12d 5198	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 5199	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 5200	Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
|- ( ( F = G /\ A = C /\ B = D ) -> ( F : A --> B <-> G : C --> D ) )