P. 55 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 5401-5500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	funeqi 5401	Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- A = B   =>    |- ( Fun A <-> Fun B )
Theorem	funeqd 5402	Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)
|- ( ph -> A = B )   =>    |- ( ph -> ( Fun A <-> Fun B ) )
Theorem	nffun 5403	Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.)
|- F/_ x F   =>    |- F/ x Fun F
Theorem	sbcfung 5404	Distribute proper substitution through the function predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
|- ( A e. V -> ( [. A / x ]. Fun F <-> Fun [_ A / x ]_ F ) )
Theorem	funeu 5405*	There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|- ( ( Fun F /\ A F B ) -> E! y A F y )
Theorem	funeu2 5406*	There is exactly one value of a function. (Contributed by NM, 3-Aug-1994.)
|- ( ( Fun F /\ <. A , B >. e. F ) -> E! y <. A , y >. e. F )
Theorem	dffun7 5407*	Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. (Enderton's definition is ambiguous because "there is only one" could mean either "there is at most one" or "there is exactly one". However, dffun8 5408 shows that it does not matter which meaning we pick.) (Contributed by NM, 4-Nov-2002.)
|- ( Fun A <-> ( Rel A /\ A. x e. dom A E* y x A y ) )
Theorem	dffun8 5408*	Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 5407. (Contributed by NM, 4-Nov-2002.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|- ( Fun A <-> ( Rel A /\ A. x e. dom A E! y x A y ) )
Theorem	dffun9 5409*	Alternate definition of a function. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.)
|- ( Fun A <-> ( Rel A /\ A. x e. dom A E* y e. ran A x A y ) )
Theorem	funfn 5410	An equivalence for the function predicate. (Contributed by NM, 13-Aug-2004.)
|- ( Fun A <-> A Fn dom A )
Theorem	funfnd 5411	A function is a function over its domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
|- ( ph -> Fun A )   =>    |- ( ph -> A Fn dom A )
Theorem	funi 5412	The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. (Contributed by NM, 30-Apr-1998.)
|- Fun _I
Theorem	nfunv 5413	The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.)
|- -. Fun _V
Theorem	funopg 5414	A Kuratowski ordered pair is a function only if its components are equal. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( ( A e. V /\ B e. W /\ Fun <. A , B >. ) -> A = B )
Theorem	funopab 5415*	A class of ordered pairs is a function when there is at most one second member for each pair. (Contributed by NM, 16-May-1995.)
|- ( Fun { <. x , y >. | ph } <-> A. x E* y ph )
Theorem	funopabeq 5416*	A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.)
|- Fun { <. x , y >. | y = A }
Theorem	funopab4 5417*	A class of ordered pairs of values in the form used by df-mpt 4195 is a function. (Contributed by NM, 17-Feb-2013.)
|- Fun { <. x , y >. | ( ph /\ y = A ) }
Theorem	funmpt 5418	A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.)
|- Fun ( x e. A |-> B )
Theorem	funmpt2 5419	Functionality of a class given by a maps-to notation. (Contributed by FL, 17-Feb-2008.) (Revised by Mario Carneiro, 31-May-2014.)
|- F = ( x e. A |-> B )   =>    |- Fun F
Theorem	funco 5420	The composition of two functions is a function. Exercise 29 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|- ( ( Fun F /\ Fun G ) -> Fun ( F o. G ) )
Theorem	funres 5421	A restriction of a function is a function. Compare Exercise 18 of [TakeutiZaring] p. 25. (Contributed by NM, 16-Aug-1994.)
|- ( Fun F -> Fun ( F |` A ) )
Theorem	funresd 5422	A restriction of a function is a function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
|- ( ph -> Fun F )   =>    |- ( ph -> Fun ( F |` A ) )
Theorem	funssres 5423	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 5424	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 5425	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	fununmo 5426*	If the union of classes is a function, there is at most one element in relation to an arbitrary element regarding one of these classes. (Contributed by AV, 18-Jul-2019.)
|- ( Fun ( F u. G ) -> E* y x F y )
Theorem	fununfun 5427	If the union of classes is a function, the classes itselves are functions. (Contributed by AV, 18-Jul-2019.)
|- ( Fun ( F u. G ) -> ( Fun F /\ Fun G ) )
Theorem	fundif 5428	A function with removed elements is still a function. (Contributed by AV, 7-Jun-2021.)
|- ( Fun F -> Fun ( F \ A ) )
Theorem	funcnvsn 5429	The converse singleton of an ordered pair is a function. This is equivalent to funsn 5432 via cnvsn 5273, 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 5430	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 5431	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 5432	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 5433	A function based on the singleton of an ordered pair. Unlike funsng 5430, 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 5434	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 5435	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 5436	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 5437	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 5438	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 5439	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 5440	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 5441	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 5442	The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.)
|- Fun (/)
Theorem	funcnvcnv 5443	The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.)
|- ( Fun A -> Fun `' `' A )
Theorem	funcnv2 5444*	A simpler equivalence for single-rooted (see funcnv 5445). (Contributed by NM, 9-Aug-2004.)
|- ( Fun `' A <-> A. y E* x x A y )
Theorem	funcnv 5445*	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 5444 for a simpler version. (Contributed by NM, 13-Aug-2004.)
|- ( Fun `' A <-> A. y e. ran A E* x x A y )
Theorem	funcnv3 5446*	A condition showing a class is single-rooted. (See funcnv 5445). (Contributed by NM, 26-May-2006.)
|- ( Fun `' A <-> A. y e. ran A E! x e. dom A x A y )
Theorem	funcnveq 5447*	Another way of expressing that a class is single-rooted. Counterpart to dffun2 5390. (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 5448*	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 5449	A single-valued relation is a function. (See fun2cnv 5448 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 5450*	Single-rootedness (see funcnv 5445) 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 5451*	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 5452*	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 5453*	The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 5445 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 5454*	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 5455	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 5456	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 5457	The converse of a restricted function. (Contributed by NM, 27-Mar-1998.)
|- ( Fun `' F -> `' ( F |` A ) = ( `' F |` ( F " A ) ) )
Theorem	cnvresid 5458	Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.)
|- `' ( _I |` A ) = ( _I |` A )
Theorem	funcnvres2 5459	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 5460	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 5461	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 5462	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 5463	One direction of imadif 5464. This direction does not require Fun `' F. (Contributed by Jim Kingdon, 25-Dec-2018.)
|- ( ( F " A ) \ ( F " B ) ) C_ ( F " ( A \ B ) )
Theorem	imadif 5464	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 5465	One direction of imain 5466. 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 5466	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 5467	Lemma for funimaexg 5468. It constitutes the interesting part of funimaexg 5468, 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 5468	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 5469	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 5470*	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 5471*	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 5469. (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 5472	Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
|- ( F = G -> ( F Fn A <-> G Fn A ) )
Theorem	fneq2 5473	Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
|- ( A = B -> ( F Fn A <-> F Fn B ) )
Theorem	fneq1d 5474	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 5475	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 5476	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 5477	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 5478	Equality inference for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
|- F = G   =>    |- ( F Fn A <-> G Fn A )
Theorem	fneq2i 5479	Equality inference for function predicate with domain. (Contributed by NM, 4-Sep-2011.)
|- A = B   =>    |- ( F Fn A <-> F Fn B )
Theorem	nffn 5480	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 5481	A function with domain is a function. (Contributed by NM, 1-Aug-1994.)
|- ( F Fn A -> Fun F )
Theorem	fnrel 5482	A function with domain is a relation. (Contributed by NM, 1-Aug-1994.)
|- ( F Fn A -> Rel F )
Theorem	fndm 5483	The domain of a function. (Contributed by NM, 2-Aug-1994.)
|- ( F Fn A -> dom F = A )
Theorem	fndmi 5484	The domain of a function. (Contributed by Wolf Lammen, 1-Jun-2024.)
|- F Fn A   =>    |- dom F = A
Theorem	fndmd 5485	The domain of a function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
|- ( ph -> F Fn A )   =>    |- ( ph -> dom F = A )
Theorem	funfni 5486	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 5487	A function has a unique domain. (Contributed by NM, 11-Aug-1994.)
|- ( ( F Fn A /\ F Fn B ) -> A = B )
Theorem	fnbr 5488	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 5489	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 5490*	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 5491*	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 5492	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 5493	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 5494	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 5495	A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.)
|- ( F Fn A -> ( F |` A ) = F )
Theorem	fnresdisj 5496	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 5497	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 5498	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 5499	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	fnssresd 5500	Restriction of a function to a subclass of its domain. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
|- ( ph -> F Fn A )   &    |- ( ph -> B C_ A )   =>    |- ( ph -> ( F |` B ) Fn B )