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	fss 5501	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 5502	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 5503	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 5504	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	fimass 5505	The image of a class under a function with domain and codomain is a subset of its codomain. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( F : A --> B -> ( F " X ) C_ B )
Theorem	fimassd 5506	The image of a class is a subset of its codomain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
|- ( ph -> F : A --> B )   =>    |- ( ph -> ( F " X ) C_ B )
Theorem	fco 5507	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	fcod 5508	Composition of two mappings. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
|- ( ph -> F : B --> C )   &    |- ( ph -> G : A --> B )   =>    |- ( ph -> ( F o. G ) : A --> C )
Theorem	fco2 5509	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 5510	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 5511	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 5512	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 5513	A mapping is a partial function. (Contributed by NM, 25-Nov-2007.)
|- ( F : A --> B -> ( F : dom F --> B /\ dom F C_ A ) )
Theorem	ffdmd 5514	The domain of a function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
|- ( ph -> F : A --> B )   =>    |- ( ph -> F : dom F --> B )
Theorem	opelf 5515	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 5516	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 5517	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	fun2d 5518	The union of functions with disjoint domains is a function, deduction version of fun2 5517. (Contributed by AV, 11-Oct-2020.) (Revised by AV, 24-Oct-2021.)
|- ( ph -> F : A --> C )   &    |- ( ph -> G : B --> C )   &    |- ( ph -> ( A i^i B ) = (/) )   =>    |- ( ph -> ( F u. G ) : ( A u. B ) --> C )
Theorem	fnfco 5519	Composition of two functions. (Contributed by NM, 22-May-2006.)
|- ( ( F Fn A /\ G : B --> A ) -> ( F o. G ) Fn B )
Theorem	fssres 5520	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 5521	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 5522	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 5523	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 5524	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 5525	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 5526	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 5527*	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 5528	The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.)
|- ( F : A --> B -> `' ( F |` A ) = ( `' F |` B ) )
Theorem	fimacnvdisj 5529	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 5530*	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 5531	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 5532*	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 5533*	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	fndmexd 5534	If a function is a set, its domain is a set. (Contributed by Rohan Ridenour, 13-May-2024.)
|- ( ph -> F e. V )   &    |- ( ph -> F Fn D )   =>    |- ( ph -> D e. _V )
Theorem	dmfex 5535	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 5536	The empty function. (Contributed by NM, 14-Aug-1999.)
|- (/) : (/) --> A
Theorem	f00 5537	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 5538	A function with empty domain is empty. (Contributed by Alexander van der Vekens, 30-Jun-2018.)
|- ( F : (/) --> X <-> F = (/) )
Theorem	f0dom0 5539	A function is empty iff it has an empty domain. (Contributed by AV, 10-Feb-2019.)
|- ( F : X --> Y -> ( X = (/) <-> F = (/) ) )
Theorem	f0rn0 5540*	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 5541	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 5542	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 5543	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 5544	Constant function with loose range. (Contributed by Stefan O'Rear, 1-Feb-2015.)
|- ( B e. C -> ( A X. { B } ) : A --> C )
Theorem	fconst6 5545	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 5546	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 5547	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 5548	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 5549	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 5550*	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 5551	A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)
|- ( F : A -1-1-> B -> F : A --> B )
Theorem	f1rn 5552	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 5553	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 5554	A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)
|- ( F : A -1-1-> B -> Fun F )
Theorem	f1rel 5555	A one-to-one onto mapping is a relation. (Contributed by NM, 8-Mar-2014.)
|- ( F : A -1-1-> B -> Rel F )
Theorem	f1dm 5556	The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.)
|- ( F : A -1-1-> B -> dom F = A )
Theorem	f1ss 5557	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 5558	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 5559	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 5560	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 5561	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 5562	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 5563	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 5564	Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
|- ( F = G -> ( F : A -onto-> B <-> G : A -onto-> B ) )
Theorem	foeq2 5565	Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
|- ( A = B -> ( F : A -onto-> C <-> F : B -onto-> C ) )
Theorem	foeq3 5566	Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
|- ( A = B -> ( F : C -onto-> A <-> F : C -onto-> B ) )
Theorem	nffo 5567	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 5568	An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)
|- ( F : A -onto-> B -> F : A --> B )
Theorem	fofun 5569	An onto mapping is a function. (Contributed by NM, 29-Mar-2008.)
|- ( F : A -onto-> B -> Fun F )
Theorem	fofn 5570	An onto mapping is a function on its domain. (Contributed by NM, 16-Dec-2008.)
|- ( F : A -onto-> B -> F Fn A )
Theorem	forn 5571	The codomain of an onto function is its range. (Contributed by NM, 3-Aug-1994.)
|- ( F : A -onto-> B -> ran F = B )
Theorem	dffo2 5572	Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.)
|- ( F : A -onto-> B <-> ( F : A --> B /\ ran F = B ) )
Theorem	foima 5573	The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)
|- ( F : A -onto-> B -> ( F " A ) = B )
Theorem	dffn4 5574	A function maps onto its range. (Contributed by NM, 10-May-1998.)
|- ( F Fn A <-> F : A -onto-> ran F )
Theorem	funforn 5575	A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.)
|- ( Fun A <-> A : dom A -onto-> ran A )
Theorem	fodmrnu 5576	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 5577	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 5578	Restriction of a function. (Contributed by NM, 4-Mar-1997.)
|- ( ( Fun F /\ A C_ dom F ) -> ( F |` A ) : A -onto-> ( F " A ) )
Theorem	foco 5579	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 5580	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 5581	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 5582	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 5583	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 5584	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 5585	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 ) )
Theorem	f1oeq123d 5586	Equality deduction for one-to-one onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
|- ( ph -> F = G )   &    |- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( F : A -1-1-onto-> C <-> G : B -1-1-onto-> D ) )
Theorem	f1oeq1d 5587	Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> F = G )   =>    |- ( ph -> ( F : A -1-1-onto-> B <-> G : A -1-1-onto-> B ) )
Theorem	f1oeq2d 5588	Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> A = B )   =>    |- ( ph -> ( F : A -1-1-onto-> C <-> F : B -1-1-onto-> C ) )
Theorem	f1oeq3d 5589	Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> A = B )   =>    |- ( ph -> ( F : C -1-1-onto-> A <-> F : C -1-1-onto-> B ) )
Theorem	nff1o 5590	Bound-variable hypothesis builder for a one-to-one onto function. (Contributed by NM, 16-May-2004.)
|- F/_ x F   &    |- F/_ x A   &    |- F/_ x B   =>    |- F/ x F : A -1-1-onto-> B
Theorem	f1of1 5591	A one-to-one onto mapping is a one-to-one mapping. (Contributed by NM, 12-Dec-2003.)
|- ( F : A -1-1-onto-> B -> F : A -1-1-> B )
Theorem	f1of 5592	A one-to-one onto mapping is a mapping. (Contributed by NM, 12-Dec-2003.)
|- ( F : A -1-1-onto-> B -> F : A --> B )
Theorem	f1ofn 5593	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 5594	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 5595	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 5596	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 5597	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 5598	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 5599	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 5600	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 ) )