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	feq123d 5401	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 5402	Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
|- ( ( F = G /\ A = C /\ B = D ) -> ( F : A --> B <-> G : C --> D ) )
Theorem	feq1i 5403	Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
|- F = G   =>    |- ( F : A --> B <-> G : A --> B )
Theorem	feq2i 5404	Equality inference for functions. (Contributed by NM, 5-Sep-2011.)
|- A = B   =>    |- ( F : A --> C <-> F : B --> C )
Theorem	feq23i 5405	Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
|- A = C   &    |- B = D   =>    |- ( F : A --> B <-> F : C --> D )
Theorem	feq23d 5406	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 5407	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
Theorem	sbcfng 5408*	Distribute proper substitution through the function predicate with a domain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
|- ( X e. V -> ( [. X / x ]. F Fn A <-> [_ X / x ]_ F Fn [_ X / x ]_ A ) )
Theorem	sbcfg 5409*	Distribute proper substitution through the function predicate with domain and codomain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
|- ( X e. V -> ( [. X / x ]. F : A --> B <-> [_ X / x ]_ F : [_ X / x ]_ A --> [_ X / x ]_ B ) )
Theorem	ffn 5410	A mapping is a function. (Contributed by NM, 2-Aug-1994.)
|- ( F : A --> B -> F Fn A )
Theorem	ffnd 5411	A mapping is a function with domain, deduction form. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> F : A --> B )   =>    |- ( ph -> F Fn A )
Theorem	dffn2 5412	Any function is a mapping into _V. (Contributed by NM, 31-Oct-1995.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|- ( F Fn A <-> F : A --> _V )
Theorem	ffun 5413	A mapping is a function. (Contributed by NM, 3-Aug-1994.)
|- ( F : A --> B -> Fun F )
Theorem	ffund 5414	A mapping is a function, deduction version. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
|- ( ph -> F : A --> B )   =>    |- ( ph -> Fun F )
Theorem	frel 5415	A mapping is a relation. (Contributed by NM, 3-Aug-1994.)
|- ( F : A --> B -> Rel F )
Theorem	fdm 5416	The domain of a mapping. (Contributed by NM, 2-Aug-1994.)
|- ( F : A --> B -> dom F = A )
Theorem	fdmd 5417	Deduction form of fdm 5416. The domain of a mapping. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
|- ( ph -> F : A --> B )   =>    |- ( ph -> dom F = A )
Theorem	fdmi 5418	The domain of a mapping. (Contributed by NM, 28-Jul-2008.)
|- F : A --> B   =>    |- dom F = A
Theorem	frn 5419	The range of a mapping. (Contributed by NM, 3-Aug-1994.)
|- ( F : A --> B -> ran F C_ B )
Theorem	frnd 5420	Deduction form of frn 5419. The range of a mapping. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
|- ( ph -> F : A --> B )   =>    |- ( ph -> ran F C_ B )
Theorem	dffn3 5421	A function maps to its range. (Contributed by NM, 1-Sep-1999.)
|- ( F Fn A <-> F : A --> ran F )
Theorem	fss 5422	Expanding the codomain of a mapping. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|- ( ( F : A --> B /\ B C_ C ) -> F : A --> C )
Theorem	fssd 5423	Expanding the codomain of a mapping, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> F : A --> B )   &    |- ( ph -> B C_ C )   =>    |- ( ph -> F : A --> C )
Theorem	fssdmd 5424	Expressing that a class is a subclass of the domain of a function expressed in maps-to notation, deduction form. (Contributed by AV, 21-Aug-2022.)
|- ( ph -> F : A --> B )   &    |- ( ph -> D C_ dom F )   =>    |- ( ph -> D C_ A )
Theorem	fssdm 5425	Expressing that a class is a subclass of the domain of a function expressed in maps-to notation, semi-deduction form. (Contributed by AV, 21-Aug-2022.)
|- D C_ dom F   &    |- ( ph -> F : A --> B )   =>    |- ( ph -> D C_ A )
Theorem	fco 5426	Composition of two mappings. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|- ( ( F : B --> C /\ G : A --> B ) -> ( F o. G ) : A --> C )
Theorem	fco2 5427	Functionality of a composition with weakened out of domain condition on the first argument. (Contributed by Stefan O'Rear, 11-Mar-2015.)
|- ( ( ( F |` B ) : B --> C /\ G : A --> B ) -> ( F o. G ) : A --> C )
Theorem	fssxp 5428	A mapping is a class of ordered pairs. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|- ( F : A --> B -> F C_ ( A X. B ) )
Theorem	fex2 5429	A function with bounded domain and codomain is a set. This version is proven without the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
|- ( ( F : A --> B /\ A e. V /\ B e. W ) -> F e. _V )
Theorem	funssxp 5430	Two ways of specifying a partial function from A to B. (Contributed by NM, 13-Nov-2007.)
|- ( ( Fun F /\ F C_ ( A X. B ) ) <-> ( F : dom F --> B /\ dom F C_ A ) )
Theorem	ffdm 5431	A mapping is a partial function. (Contributed by NM, 25-Nov-2007.)
|- ( F : A --> B -> ( F : dom F --> B /\ dom F C_ A ) )
Theorem	opelf 5432	The members of an ordered pair element of a mapping belong to the mapping's domain and codomain. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( ( F : A --> B /\ <. C , D >. e. F ) -> ( C e. A /\ D e. B ) )
Theorem	fun 5433	The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
|- ( ( ( F : A --> C /\ G : B --> D ) /\ ( A i^i B ) = (/) ) -> ( F u. G ) : ( A u. B ) --> ( C u. D ) )
Theorem	fun2 5434	The union of two functions with disjoint domains. (Contributed by Mario Carneiro, 12-Mar-2015.)
|- ( ( ( F : A --> C /\ G : B --> C ) /\ ( A i^i B ) = (/) ) -> ( F u. G ) : ( A u. B ) --> C )
Theorem	fnfco 5435	Composition of two functions. (Contributed by NM, 22-May-2006.)
|- ( ( F Fn A /\ G : B --> A ) -> ( F o. G ) Fn B )
Theorem	fssres 5436	Restriction of a function with a subclass of its domain. (Contributed by NM, 23-Sep-2004.)
|- ( ( F : A --> B /\ C C_ A ) -> ( F |` C ) : C --> B )
Theorem	fssresd 5437	Restriction of a function with a subclass of its domain, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> F : A --> B )   &    |- ( ph -> C C_ A )   =>    |- ( ph -> ( F |` C ) : C --> B )
Theorem	fssres2 5438	Restriction of a restricted function with a subclass of its domain. (Contributed by NM, 21-Jul-2005.)
|- ( ( ( F |` A ) : A --> B /\ C C_ A ) -> ( F |` C ) : C --> B )
Theorem	fresin 5439	An identity for the mapping relationship under restriction. (Contributed by Scott Fenton, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 26-May-2016.)
|- ( F : A --> B -> ( F |` X ) : ( A i^i X ) --> B )
Theorem	resasplitss 5440	If two functions agree on their common domain, their union contains a union of three functions with pairwise disjoint domains. If we assumed the law of the excluded middle, this would be equality rather than subset. (Contributed by Jim Kingdon, 28-Dec-2018.)
|- ( ( F Fn A /\ G Fn B /\ ( F |` ( A i^i B ) ) = ( G |` ( A i^i B ) ) ) -> ( ( F |` ( A i^i B ) ) u. ( ( F |` ( A \ B ) ) u. ( G |` ( B \ A ) ) ) ) C_ ( F u. G ) )
Theorem	fcoi1 5441	Composition of a mapping and restricted identity. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|- ( F : A --> B -> ( F o. ( _I |` A ) ) = F )
Theorem	fcoi2 5442	Composition of restricted identity and a mapping. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|- ( F : A --> B -> ( ( _I |` B ) o. F ) = F )
Theorem	feu 5443*	There is exactly one value of a function in its codomain. (Contributed by NM, 10-Dec-2003.)
|- ( ( F : A --> B /\ C e. A ) -> E! y e. B <. C , y >. e. F )
Theorem	fcnvres 5444	The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.)
|- ( F : A --> B -> `' ( F |` A ) = ( `' F |` B ) )
Theorem	fimacnvdisj 5445	The preimage of a class disjoint with a mapping's codomain is empty. (Contributed by FL, 24-Jan-2007.)
|- ( ( F : A --> B /\ ( B i^i C ) = (/) ) -> ( `' F " C ) = (/) )
Theorem	fintm 5446*	Function into an intersection. (Contributed by Jim Kingdon, 28-Dec-2018.)
|- E. x x e. B   =>    |- ( F : A --> |^| B <-> A. x e. B F : A --> x )
Theorem	fin 5447	Mapping into an intersection. (Contributed by NM, 14-Sep-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|- ( F : A --> ( B i^i C ) <-> ( F : A --> B /\ F : A --> C ) )
Theorem	fabexg 5448*	Existence of a set of functions. (Contributed by Paul Chapman, 25-Feb-2008.)
|- F = { x | ( x : A --> B /\ ph ) }   =>    |- ( ( A e. C /\ B e. D ) -> F e. _V )
Theorem	fabex 5449*	Existence of a set of functions. (Contributed by NM, 3-Dec-2007.)
|- A e. _V   &    |- B e. _V   &    |- F = { x | ( x : A --> B /\ ph ) }   =>    |- F e. _V
Theorem	dmfex 5450	If a mapping is a set, its domain is a set. (Contributed by NM, 27-Aug-2006.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|- ( ( F e. C /\ F : A --> B ) -> A e. _V )
Theorem	f0 5451	The empty function. (Contributed by NM, 14-Aug-1999.)
|- (/) : (/) --> A
Theorem	f00 5452	A class is a function with empty codomain iff it and its domain are empty. (Contributed by NM, 10-Dec-2003.)
|- ( F : A --> (/) <-> ( F = (/) /\ A = (/) ) )
Theorem	f0bi 5453	A function with empty domain is empty. (Contributed by Alexander van der Vekens, 30-Jun-2018.)
|- ( F : (/) --> X <-> F = (/) )
Theorem	f0dom0 5454	A function is empty iff it has an empty domain. (Contributed by AV, 10-Feb-2019.)
|- ( F : X --> Y -> ( X = (/) <-> F = (/) ) )
Theorem	f0rn0 5455*	If there is no element in the range of a function, its domain must be empty. (Contributed by Alexander van der Vekens, 12-Jul-2018.)
|- ( ( E : X --> Y /\ -. E. y e. Y y e. ran E ) -> X = (/) )
Theorem	fconst 5456	A cross product with a singleton is a constant function. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|- B e. _V   =>    |- ( A X. { B } ) : A --> { B }
Theorem	fconstg 5457	A cross product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.)
|- ( B e. V -> ( A X. { B } ) : A --> { B } )
Theorem	fnconstg 5458	A cross product with a singleton is a constant function. (Contributed by NM, 24-Jul-2014.)
|- ( B e. V -> ( A X. { B } ) Fn A )
Theorem	fconst6g 5459	Constant function with loose range. (Contributed by Stefan O'Rear, 1-Feb-2015.)
|- ( B e. C -> ( A X. { B } ) : A --> C )
Theorem	fconst6 5460	A constant function as a mapping. (Contributed by Jeff Madsen, 30-Nov-2009.) (Revised by Mario Carneiro, 22-Apr-2015.)
|- B e. C   =>    |- ( A X. { B } ) : A --> C
Theorem	f1eq1 5461	Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
|- ( F = G -> ( F : A -1-1-> B <-> G : A -1-1-> B ) )
Theorem	f1eq2 5462	Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
|- ( A = B -> ( F : A -1-1-> C <-> F : B -1-1-> C ) )
Theorem	f1eq3 5463	Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
|- ( A = B -> ( F : C -1-1-> A <-> F : C -1-1-> B ) )
Theorem	nff1 5464	Bound-variable hypothesis builder for a one-to-one function. (Contributed by NM, 16-May-2004.)
|- F/_ x F   &    |- F/_ x A   &    |- F/_ x B   =>    |- F/ x F : A -1-1-> B
Theorem	dff12 5465*	Alternate definition of a one-to-one function. (Contributed by NM, 31-Dec-1996.)
|- ( F : A -1-1-> B <-> ( F : A --> B /\ A. y E* x x F y ) )
Theorem	f1f 5466	A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)
|- ( F : A -1-1-> B -> F : A --> B )
Theorem	f1rn 5467	The range of a one-to-one mapping. (Contributed by BJ, 6-Jul-2022.)
|- ( F : A -1-1-> B -> ran F C_ B )
Theorem	f1fn 5468	A one-to-one mapping is a function on its domain. (Contributed by NM, 8-Mar-2014.)
|- ( F : A -1-1-> B -> F Fn A )
Theorem	f1fun 5469	A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)
|- ( F : A -1-1-> B -> Fun F )
Theorem	f1rel 5470	A one-to-one onto mapping is a relation. (Contributed by NM, 8-Mar-2014.)
|- ( F : A -1-1-> B -> Rel F )
Theorem	f1dm 5471	The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.)
|- ( F : A -1-1-> B -> dom F = A )
Theorem	f1ss 5472	A function that is one-to-one is also one-to-one on some superset of its range. (Contributed by Mario Carneiro, 12-Jan-2013.)
|- ( ( F : A -1-1-> B /\ B C_ C ) -> F : A -1-1-> C )
Theorem	f1ssr 5473	Combine a one-to-one function with a restriction on the domain. (Contributed by Stefan O'Rear, 20-Feb-2015.)
|- ( ( F : A -1-1-> B /\ ran F C_ C ) -> F : A -1-1-> C )
Theorem	f1ff1 5474	If a function is one-to-one from A to B and is also a function from A to C, then it is a one-to-one function from A to C. (Contributed by BJ, 4-Jul-2022.)
|- ( ( F : A -1-1-> B /\ F : A --> C ) -> F : A -1-1-> C )
Theorem	f1ssres 5475	A function that is one-to-one is also one-to-one on any subclass of its domain. (Contributed by Mario Carneiro, 17-Jan-2015.)
|- ( ( F : A -1-1-> B /\ C C_ A ) -> ( F |` C ) : C -1-1-> B )
Theorem	f1resf1 5476	The restriction of an injective function is injective. (Contributed by AV, 28-Jun-2022.)
|- ( ( ( F : A -1-1-> B /\ C C_ A ) /\ ( F |` C ) : C --> D ) -> ( F |` C ) : C -1-1-> D )
Theorem	f1cnvcnv 5477	Two ways to express that a set A (not necessarily a function) is one-to-one. Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un2 (A)" for one-to-one. We do not introduce a separate notation since we rarely use it. (Contributed by NM, 13-Aug-2004.)
|- ( `' `' A : dom A -1-1-> _V <-> ( Fun `' A /\ Fun `' `' A ) )
Theorem	f1co 5478	Composition of one-to-one functions. Exercise 30 of [TakeutiZaring] p. 25. (Contributed by NM, 28-May-1998.)
|- ( ( F : B -1-1-> C /\ G : A -1-1-> B ) -> ( F o. G ) : A -1-1-> C )
Theorem	foeq1 5479	Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
|- ( F = G -> ( F : A -onto-> B <-> G : A -onto-> B ) )
Theorem	foeq2 5480	Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
|- ( A = B -> ( F : A -onto-> C <-> F : B -onto-> C ) )
Theorem	foeq3 5481	Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
|- ( A = B -> ( F : C -onto-> A <-> F : C -onto-> B ) )
Theorem	nffo 5482	Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.)
|- F/_ x F   &    |- F/_ x A   &    |- F/_ x B   =>    |- F/ x F : A -onto-> B
Theorem	fof 5483	An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)
|- ( F : A -onto-> B -> F : A --> B )
Theorem	fofun 5484	An onto mapping is a function. (Contributed by NM, 29-Mar-2008.)
|- ( F : A -onto-> B -> Fun F )
Theorem	fofn 5485	An onto mapping is a function on its domain. (Contributed by NM, 16-Dec-2008.)
|- ( F : A -onto-> B -> F Fn A )
Theorem	forn 5486	The codomain of an onto function is its range. (Contributed by NM, 3-Aug-1994.)
|- ( F : A -onto-> B -> ran F = B )
Theorem	dffo2 5487	Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.)
|- ( F : A -onto-> B <-> ( F : A --> B /\ ran F = B ) )
Theorem	foima 5488	The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)
|- ( F : A -onto-> B -> ( F " A ) = B )
Theorem	dffn4 5489	A function maps onto its range. (Contributed by NM, 10-May-1998.)
|- ( F Fn A <-> F : A -onto-> ran F )
Theorem	funforn 5490	A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.)
|- ( Fun A <-> A : dom A -onto-> ran A )
Theorem	fodmrnu 5491	An onto function has unique domain and range. (Contributed by NM, 5-Nov-2006.)
|- ( ( F : A -onto-> B /\ F : C -onto-> D ) -> ( A = C /\ B = D ) )
Theorem	fimadmfo 5492	A function is a function onto the image of its domain. (Contributed by AV, 1-Dec-2022.)
|- ( F : A --> B -> F : A -onto-> ( F " A ) )
Theorem	fores 5493	Restriction of a function. (Contributed by NM, 4-Mar-1997.)
|- ( ( Fun F /\ A C_ dom F ) -> ( F |` A ) : A -onto-> ( F " A ) )
Theorem	foco 5494	Composition of onto functions. (Contributed by NM, 22-Mar-2006.)
|- ( ( F : B -onto-> C /\ G : A -onto-> B ) -> ( F o. G ) : A -onto-> C )
Theorem	f1oeq1 5495	Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
|- ( F = G -> ( F : A -1-1-onto-> B <-> G : A -1-1-onto-> B ) )
Theorem	f1oeq2 5496	Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
|- ( A = B -> ( F : A -1-1-onto-> C <-> F : B -1-1-onto-> C ) )
Theorem	f1oeq3 5497	Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
|- ( A = B -> ( F : C -1-1-onto-> A <-> F : C -1-1-onto-> B ) )
Theorem	f1oeq23 5498	Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012.)
|- ( ( A = B /\ C = D ) -> ( F : A -1-1-onto-> C <-> F : B -1-1-onto-> D ) )
Theorem	f1eq123d 5499	Equality deduction for one-to-one functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
|- ( ph -> F = G )   &    |- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( F : A -1-1-> C <-> G : B -1-1-> D ) )
Theorem	foeq123d 5500	Equality deduction for onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
|- ( ph -> F = G )   &    |- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( F : A -onto-> C <-> G : B -onto-> D ) )