P. 51 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 5001-5100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fnbr 5001	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 5002	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 5003*	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 5004*	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 5005	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 5006	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 5007	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 5008	A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.)
|- ( F Fn A -> ( F |` A ) = F )
Theorem	fnresdisj 5009	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 5010	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 5011	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 5012	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 5013	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 5014	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 5015*	An equivalence for functionality of a restriction. Compare dffun8 4929. (Contributed by Mario Carneiro, 20-May-2015.)
|- ( ( F |` A ) Fn A <-> A. x e. A E! y x F y )
Theorem	fnresi 5016	Functionality and domain of restricted identity. (Contributed by NM, 27-Aug-2004.)
|- ( _I |` A ) Fn A
Theorem	fnima 5017	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 5018	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 5019	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 5020	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 5021	Alternate definition for the "maps to" notation df-mpt 3820. (Contributed by Mario Carneiro, 30-Dec-2016.)
|- ( x e. A |-> B ) = U_ x e. A ( { x } X. { B } )
Theorem	fnopabg 5022*	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 5023*	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 5024*	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 5025*	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 5026	A mapping operation with empty domain. (Contributed by Mario Carneiro, 28-Dec-2014.)
|- ( x e. (/) |-> A ) = (/)
Theorem	fnmpti 5027*	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 5028*	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	mptun 5029	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 5030	Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
|- ( F = G -> ( F : A --> B <-> G : A --> B ) )
Theorem	feq2 5031	Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
|- ( A = B -> ( F : A --> C <-> F : B --> C ) )
Theorem	feq3 5032	Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
|- ( A = B -> ( F : C --> A <-> F : C --> B ) )
Theorem	feq23 5033	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 5034	Equality deduction for functions. (Contributed by NM, 19-Feb-2008.)
|- ( ph -> F = G )   =>    |- ( ph -> ( F : A --> B <-> G : A --> B ) )
Theorem	feq2d 5035	Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ph -> A = B )   =>    |- ( ph -> ( F : A --> C <-> F : B --> C ) )
Theorem	feq12d 5036	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 5037	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 5038	Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
|- ( ( F = G /\ A = C /\ B = D ) -> ( F : A --> B <-> G : C --> D ) )
Theorem	feq1i 5039	Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
|- F = G   =>    |- ( F : A --> B <-> G : A --> B )
Theorem	feq2i 5040	Equality inference for functions. (Contributed by NM, 5-Sep-2011.)
|- A = B   =>    |- ( F : A --> C <-> F : B --> C )
Theorem	feq23i 5041	Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
|- A = C   &    |- B = D   =>    |- ( F : A --> B <-> F : C --> D )
Theorem	feq23d 5042	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 5043	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 5044*	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 5045*	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 5046	A mapping is a function. (Contributed by NM, 2-Aug-1994.)
|- ( F : A --> B -> F Fn A )
Theorem	dffn2 5047	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 5048	A mapping is a function. (Contributed by NM, 3-Aug-1994.)
|- ( F : A --> B -> Fun F )
Theorem	frel 5049	A mapping is a relation. (Contributed by NM, 3-Aug-1994.)
|- ( F : A --> B -> Rel F )
Theorem	fdm 5050	The domain of a mapping. (Contributed by NM, 2-Aug-1994.)
|- ( F : A --> B -> dom F = A )
Theorem	fdmi 5051	The domain of a mapping. (Contributed by NM, 28-Jul-2008.)
|- F : A --> B   =>    |- dom F = A
Theorem	frn 5052	The range of a mapping. (Contributed by NM, 3-Aug-1994.)
|- ( F : A --> B -> ran F C_ B )
Theorem	dffn3 5053	A function maps to its range. (Contributed by NM, 1-Sep-1999.)
|- ( F Fn A <-> F : A --> ran F )
Theorem	fss 5054	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 5055	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	fco 5056	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 5057	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 5058	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 5059	A function with bounded domain and range 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 5060	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 5061	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 5062	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 5063	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 5064	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 5065	Composition of two functions. (Contributed by NM, 22-May-2006.)
|- ( ( F Fn A /\ G : B --> A ) -> ( F o. G ) Fn B )
Theorem	fssres 5066	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	fssres2 5067	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 5068	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 5069	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 5070	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 5071	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 5072*	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 5073	The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.)
|- ( F : A --> B -> `' ( F |` A ) = ( `' F |` B ) )
Theorem	fimacnvdisj 5074	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 5075*	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 5076	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 5077*	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 5078*	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 5079	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 5080	The empty function. (Contributed by NM, 14-Aug-1999.)
|- (/) : (/) --> A
Theorem	f00 5081	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	fconst 5082	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 5083	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 5084	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 5085	Constant function with loose range. (Contributed by Stefan O'Rear, 1-Feb-2015.)
|- ( B e. C -> ( A X. { B } ) : A --> C )
Theorem	fconst6 5086	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 5087	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 5088	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 5089	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 5090	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 5091*	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 5092	A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)
|- ( F : A -1-1-> B -> F : A --> B )
Theorem	f1fn 5093	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 5094	A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)
|- ( F : A -1-1-> B -> Fun F )
Theorem	f1rel 5095	A one-to-one onto mapping is a relation. (Contributed by NM, 8-Mar-2014.)
|- ( F : A -1-1-> B -> Rel F )
Theorem	f1dm 5096	The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.)
|- ( F : A -1-1-> B -> dom F = A )
Theorem	f1ss 5097	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 5098	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	f1ssres 5099	A function that is one-to-one is also one-to-one on some aubset 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	f1cnvcnv 5100	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 ) )