P. 61 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 6001-6100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fnresdisj 6001	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 6002	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 6003	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 6004	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 6005	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 6006	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 6007*	An equivalence for functionality of a restriction. Compare dffun8 5916. (Contributed by Mario Carneiro, 20-May-2015.)
|- ( ( F |` A ) Fn A <-> A. x e. A E! y x F y )
Theorem	fnresi 6008	Functionality and domain of restricted identity. (Contributed by NM, 27-Aug-2004.)
|- ( _I |` A ) Fn A
Theorem	idssxp 6009	A diagonal set as a subset of a Cartesian product. (Contributed by Thierry Arnoux, 29-Dec-2019.)
|- ( _I |` A ) C_ ( A X. A )
Theorem	fnima 6010	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 6011	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 6012	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 6013	Images under a function never map nonempty sets to empty sets. EDITORIAL: usable in fnwe2lem2 37621. (Contributed by Stefan O'Rear, 21-Jan-2015.)
|- ( ( F Fn A /\ B C_ A ) -> ( ( F " B ) = (/) <-> B = (/) ) )
Theorem	dfmpt3 6014	Alternate definition for the "maps to" notation df-mpt 4730. (Contributed by Mario Carneiro, 30-Dec-2016.)
|- ( x e. A |-> B ) = U_ x e. A ( { x } X. { B } )
Theorem	mptfnf 6015	The maps-to notation defines a function with domain. (Contributed by Scott Fenton, 21-Mar-2011.) (Revised by Thierry Arnoux, 10-May-2017.)
|- F/_ x A   =>    |- ( A. x e. A B e. _V <-> ( x e. A |-> B ) Fn A )
Theorem	fnmptf 6016	The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.) (Revised by Thierry Arnoux, 10-May-2017.)
|- F/_ x A   =>    |- ( A. x e. A B e. V -> ( x e. A |-> B ) Fn A )
Theorem	fnopabg 6017*	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 6018*	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 6019*	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 6020*	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 6021	A mapping operation with empty domain. (Contributed by Mario Carneiro, 28-Dec-2014.)
|- ( x e. (/) |-> A ) = (/)
Theorem	fnmpti 6022*	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 6023*	Domain of the mapping operation. (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 6024*	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 6025	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 6026	Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
|- ( F = G -> ( F : A --> B <-> G : A --> B ) )
Theorem	feq2 6027	Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
|- ( A = B -> ( F : A --> C <-> F : B --> C ) )
Theorem	feq3 6028	Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
|- ( A = B -> ( F : C --> A <-> F : C --> B ) )
Theorem	feq23 6029	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 6030	Equality deduction for functions. (Contributed by NM, 19-Feb-2008.)
|- ( ph -> F = G )   =>    |- ( ph -> ( F : A --> B <-> G : A --> B ) )
Theorem	feq2d 6031	Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ph -> A = B )   =>    |- ( ph -> ( F : A --> C <-> F : B --> C ) )
Theorem	feq3d 6032	Equality deduction for functions. (Contributed by AV, 1-Jan-2020.)
|- ( ph -> A = B )   =>    |- ( ph -> ( F : X --> A <-> F : X --> B ) )
Theorem	feq12d 6033	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 6034	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 6035	Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
|- ( ( F = G /\ A = C /\ B = D ) -> ( F : A --> B <-> G : C --> D ) )
Theorem	feq1i 6036	Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
|- F = G   =>    |- ( F : A --> B <-> G : A --> B )
Theorem	feq2i 6037	Equality inference for functions. (Contributed by NM, 5-Sep-2011.)
|- A = B   =>    |- ( F : A --> C <-> F : B --> C )
Theorem	feq12i 6038	Equality inference for functions. (Contributed by AV, 7-Feb-2021.)
|- F = G   &    |- A = B   =>    |- ( F : A --> C <-> G : B --> C )
Theorem	feq23i 6039	Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
|- A = C   &    |- B = D   =>    |- ( F : A --> B <-> F : C --> D )
Theorem	feq23d 6040	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 6041	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 6042*	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 6043*	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	elimf 6044	Eliminate a mapping hypothesis for the weak deduction theorem dedth 4139, when a special case G : A --> B is provable, in order to convert F : A --> B from a hypothesis to an antecedent. (Contributed by NM, 24-Aug-2006.)
|- G : A --> B   =>    |- if ( F : A --> B , F , G ) : A --> B
Theorem	ffn 6045	A mapping is a function with domain. (Contributed by NM, 2-Aug-1994.)
|- ( F : A --> B -> F Fn A )
Theorem	ffnd 6046	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 6047	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 6048	A mapping is a function. (Contributed by NM, 3-Aug-1994.)
|- ( F : A --> B -> Fun F )
Theorem	ffund 6049	A mapping is a function, deduction version. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
|- ( ph -> F : A --> B )   =>    |- ( ph -> Fun F )
Theorem	frel 6050	A mapping is a relation. (Contributed by NM, 3-Aug-1994.)
|- ( F : A --> B -> Rel F )
Theorem	fdm 6051	The domain of a mapping. (Contributed by NM, 2-Aug-1994.)
|- ( F : A --> B -> dom F = A )
Theorem	fdmi 6052	The domain of a mapping. (Contributed by NM, 28-Jul-2008.)
|- F : A --> B   =>    |- dom F = A
Theorem	frn 6053	The range of a mapping. (Contributed by NM, 3-Aug-1994.)
|- ( F : A --> B -> ran F C_ B )
Theorem	dffn3 6054	A function maps to its range. (Contributed by NM, 1-Sep-1999.)
|- ( F Fn A <-> F : A --> ran F )
Theorem	ffrn 6055	A function maps to its range. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
|- ( F : A --> B -> F : A --> ran F )
Theorem	fss 6056	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 6057	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 6058	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 6059	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 6060	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	funssxp 6061	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 6062	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 6063	The domain of a function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
|- ( ph -> F : A --> B )   =>    |- ( ph -> F : dom F --> B )
Theorem	fdmrn 6064	A different way to write F is a function. (Contributed by Thierry Arnoux, 7-Dec-2016.)
|- ( Fun F <-> F : dom F --> ran F )
Theorem	opelf 6065	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 6066	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 6067	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 6068	The union of functions with disjoint domains is a function, deduction version of fun2 6067. (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 6069	Composition of two functions. (Contributed by NM, 22-May-2006.)
|- ( ( F Fn A /\ G : B --> A ) -> ( F o. G ) Fn B )
Theorem	fssres 6070	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 6071	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 6072	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 6073	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	resasplit 6074	If two functions agree on their common domain, express their union as a union of three functions with pairwise disjoint domains. (Contributed by Stefan O'Rear, 9-Oct-2014.)
|- ( ( F Fn A /\ G Fn B /\ ( F |` ( A i^i B ) ) = ( G |` ( A i^i B ) ) ) -> ( F u. G ) = ( ( F |` ( A i^i B ) ) u. ( ( F |` ( A \ B ) ) u. ( G |` ( B \ A ) ) ) ) )
Theorem	fresaun 6075	The union of two functions which agree on their common domain is a function. (Contributed by Stefan O'Rear, 9-Oct-2014.)
|- ( ( F : A --> C /\ G : B --> C /\ ( F |` ( A i^i B ) ) = ( G |` ( A i^i B ) ) ) -> ( F u. G ) : ( A u. B ) --> C )
Theorem	fresaunres2 6076	From the union of two functions that agree on the domain overlap, either component can be recovered by restriction. (Contributed by Stefan O'Rear, 9-Oct-2014.)
|- ( ( F : A --> C /\ G : B --> C /\ ( F |` ( A i^i B ) ) = ( G |` ( A i^i B ) ) ) -> ( ( F u. G ) |` B ) = G )
Theorem	fresaunres1 6077	From the union of two functions that agree on the domain overlap, either component can be recovered by restriction. (Contributed by Mario Carneiro, 16-Feb-2015.)
|- ( ( F : A --> C /\ G : B --> C /\ ( F |` ( A i^i B ) ) = ( G |` ( A i^i B ) ) ) -> ( ( F u. G ) |` A ) = F )
Theorem	fcoi1 6078	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 6079	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 6080*	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	fimass 6081	The image of a class is a subset of its codomain. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( F : A --> B -> ( F " X ) C_ B )
Theorem	fcnvres 6082	The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.)
|- ( F : A --> B -> `' ( F |` A ) = ( `' F |` B ) )
Theorem	fimacnvdisj 6083	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	fint 6084*	Function into an intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|- B =/= (/)   =>    |- ( F : A --> |^| B <-> A. x e. B F : A --> x )
Theorem	fin 6085	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	f0 6086	The empty function. (Contributed by NM, 14-Aug-1999.)
|- (/) : (/) --> A
Theorem	f00 6087	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 6088	A function with empty domain is empty. (Contributed by Alexander van der Vekens, 30-Jun-2018.)
|- ( F : (/) --> X <-> F = (/) )
Theorem	f0dom0 6089	A function is empty iff it has an empty domain. (Contributed by AV, 10-Feb-2019.)
|- ( F : X --> Y -> ( X = (/) <-> F = (/) ) )
Theorem	f0rn0 6090*	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 6091	A Cartesian 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 6092	A Cartesian product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.)
|- ( B e. V -> ( A X. { B } ) : A --> { B } )
Theorem	fnconstg 6093	A Cartesian product with a singleton is a constant function. (Contributed by NM, 24-Jul-2014.)
|- ( B e. V -> ( A X. { B } ) Fn A )
Theorem	fconst6g 6094	Constant function with loose range. (Contributed by Stefan O'Rear, 1-Feb-2015.)
|- ( B e. C -> ( A X. { B } ) : A --> C )
Theorem	fconst6 6095	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 6096	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 6097	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 6098	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 6099	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 6100*	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 ) )