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	imainlem 5401	One direction of imain 5402. 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 5402	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 5403	Lemma for funimaexg 5404. It constitutes the interesting part of funimaexg 5404, 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 5404	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 5405	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 5406*	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 5407*	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 5405. (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 5408	Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
|- ( F = G -> ( F Fn A <-> G Fn A ) )
Theorem	fneq2 5409	Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
|- ( A = B -> ( F Fn A <-> F Fn B ) )
Theorem	fneq1d 5410	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 5411	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 5412	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 5413	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 5414	Equality inference for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
|- F = G   =>    |- ( F Fn A <-> G Fn A )
Theorem	fneq2i 5415	Equality inference for function predicate with domain. (Contributed by NM, 4-Sep-2011.)
|- A = B   =>    |- ( F Fn A <-> F Fn B )
Theorem	nffn 5416	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 5417	A function with domain is a function. (Contributed by NM, 1-Aug-1994.)
|- ( F Fn A -> Fun F )
Theorem	fnrel 5418	A function with domain is a relation. (Contributed by NM, 1-Aug-1994.)
|- ( F Fn A -> Rel F )
Theorem	fndm 5419	The domain of a function. (Contributed by NM, 2-Aug-1994.)
|- ( F Fn A -> dom F = A )
Theorem	fndmi 5420	The domain of a function. (Contributed by Wolf Lammen, 1-Jun-2024.)
|- F Fn A   =>    |- dom F = A
Theorem	fndmd 5421	The domain of a function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
|- ( ph -> F Fn A )   =>    |- ( ph -> dom F = A )
Theorem	funfni 5422	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 5423	A function has a unique domain. (Contributed by NM, 11-Aug-1994.)
|- ( ( F Fn A /\ F Fn B ) -> A = B )
Theorem	fnbr 5424	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 5425	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 5426*	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 5427*	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 5428	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 5429	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 5430	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 5431	A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.)
|- ( F Fn A -> ( F |` A ) = F )
Theorem	fnresdisj 5432	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 5433	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 5434	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 5435	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 5436	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 )
Theorem	fnresin1 5437	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 5438	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 5439*	An equivalence for functionality of a restriction. Compare dffun8 5345. (Contributed by Mario Carneiro, 20-May-2015.)
|- ( ( F |` A ) Fn A <-> A. x e. A E! y x F y )
Theorem	fnresi 5440	Functionality and domain of restricted identity. (Contributed by NM, 27-Aug-2004.)
|- ( _I |` A ) Fn A
Theorem	fnima 5441	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 5442	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 5443	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 5444	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 5445	Alternate definition for the maps-to notation df-mpt 4146. (Contributed by Mario Carneiro, 30-Dec-2016.)
|- ( x e. A |-> B ) = U_ x e. A ( { x } X. { B } )
Theorem	fnopabg 5446*	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 5447*	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 5448*	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 5449*	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 5450	A mapping operation with empty domain. (Contributed by Mario Carneiro, 28-Dec-2014.)
|- ( x e. (/) |-> A ) = (/)
Theorem	fnmpti 5451*	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 5452*	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	dmmptd 5453*	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 5454	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 5455	Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
|- ( F = G -> ( F : A --> B <-> G : A --> B ) )
Theorem	feq2 5456	Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
|- ( A = B -> ( F : A --> C <-> F : B --> C ) )
Theorem	feq3 5457	Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
|- ( A = B -> ( F : C --> A <-> F : C --> B ) )
Theorem	feq23 5458	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 5459	Equality deduction for functions. (Contributed by NM, 19-Feb-2008.)
|- ( ph -> F = G )   =>    |- ( ph -> ( F : A --> B <-> G : A --> B ) )
Theorem	feq2d 5460	Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ph -> A = B )   =>    |- ( ph -> ( F : A --> C <-> F : B --> C ) )
Theorem	feq3d 5461	Equality deduction for functions. (Contributed by AV, 1-Jan-2020.)
|- ( ph -> A = B )   =>    |- ( ph -> ( F : X --> A <-> F : X --> B ) )
Theorem	feq12d 5462	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 5463	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 5464	Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
|- ( ( F = G /\ A = C /\ B = D ) -> ( F : A --> B <-> G : C --> D ) )
Theorem	feq1i 5465	Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
|- F = G   =>    |- ( F : A --> B <-> G : A --> B )
Theorem	feq2i 5466	Equality inference for functions. (Contributed by NM, 5-Sep-2011.)
|- A = B   =>    |- ( F : A --> C <-> F : B --> C )
Theorem	feq23i 5467	Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
|- A = C   &    |- B = D   =>    |- ( F : A --> B <-> F : C --> D )
Theorem	feq23d 5468	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 5469	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 5470*	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 5471*	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 5472	A mapping is a function. (Contributed by NM, 2-Aug-1994.)
|- ( F : A --> B -> F Fn A )
Theorem	ffnd 5473	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 5474	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 5475	A mapping is a function. (Contributed by NM, 3-Aug-1994.)
|- ( F : A --> B -> Fun F )
Theorem	ffund 5476	A mapping is a function, deduction version. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
|- ( ph -> F : A --> B )   =>    |- ( ph -> Fun F )
Theorem	frel 5477	A mapping is a relation. (Contributed by NM, 3-Aug-1994.)
|- ( F : A --> B -> Rel F )
Theorem	fdm 5478	The domain of a mapping. (Contributed by NM, 2-Aug-1994.)
|- ( F : A --> B -> dom F = A )
Theorem	fdmd 5479	Deduction form of fdm 5478. The domain of a mapping. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
|- ( ph -> F : A --> B )   =>    |- ( ph -> dom F = A )
Theorem	fdmi 5480	The domain of a mapping. (Contributed by NM, 28-Jul-2008.)
|- F : A --> B   =>    |- dom F = A
Theorem	frn 5481	The range of a mapping. (Contributed by NM, 3-Aug-1994.)
|- ( F : A --> B -> ran F C_ B )
Theorem	frnd 5482	Deduction form of frn 5481. The range of a mapping. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
|- ( ph -> F : A --> B )   =>    |- ( ph -> ran F C_ B )
Theorem	dffn3 5483	A function maps to its range. (Contributed by NM, 1-Sep-1999.)
|- ( F Fn A <-> F : A --> ran F )
Theorem	fss 5484	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 5485	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 5486	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 5487	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 5488	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 5489	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 5490	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 5491	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 5492	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 5493	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 5494	The domain of a function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
|- ( ph -> F : A --> B )   =>    |- ( ph -> F : dom F --> B )
Theorem	opelf 5495	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 5496	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 5497	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 5498	The union of functions with disjoint domains is a function, deduction version of fun2 5497. (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 5499	Composition of two functions. (Contributed by NM, 22-May-2006.)
|- ( ( F Fn A /\ G : B --> A ) -> ( F o. G ) Fn B )
Theorem	fssres 5500	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 )