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	fcnvres 5401	The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.)
|- ( F : A --> B -> `' ( F |` A ) = ( `' F |` B ) )
Theorem	fimacnvdisj 5402	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 5403*	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 5404	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 5405*	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 5406*	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 5407	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 5408	The empty function. (Contributed by NM, 14-Aug-1999.)
|- (/) : (/) --> A
Theorem	f00 5409	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 5410	A function with empty domain is empty. (Contributed by Alexander van der Vekens, 30-Jun-2018.)
|- ( F : (/) --> X <-> F = (/) )
Theorem	f0dom0 5411	A function is empty iff it has an empty domain. (Contributed by AV, 10-Feb-2019.)
|- ( F : X --> Y -> ( X = (/) <-> F = (/) ) )
Theorem	f0rn0 5412*	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 5413	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 5414	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 5415	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 5416	Constant function with loose range. (Contributed by Stefan O'Rear, 1-Feb-2015.)
|- ( B e. C -> ( A X. { B } ) : A --> C )
Theorem	fconst6 5417	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 5418	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 5419	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 5420	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 5421	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 5422*	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 5423	A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)
|- ( F : A -1-1-> B -> F : A --> B )
Theorem	f1rn 5424	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 5425	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 5426	A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)
|- ( F : A -1-1-> B -> Fun F )
Theorem	f1rel 5427	A one-to-one onto mapping is a relation. (Contributed by NM, 8-Mar-2014.)
|- ( F : A -1-1-> B -> Rel F )
Theorem	f1dm 5428	The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.)
|- ( F : A -1-1-> B -> dom F = A )
Theorem	f1ss 5429	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 5430	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 5431	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 5432	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 5433	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 5434	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 5435	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 5436	Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
|- ( F = G -> ( F : A -onto-> B <-> G : A -onto-> B ) )
Theorem	foeq2 5437	Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
|- ( A = B -> ( F : A -onto-> C <-> F : B -onto-> C ) )
Theorem	foeq3 5438	Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
|- ( A = B -> ( F : C -onto-> A <-> F : C -onto-> B ) )
Theorem	nffo 5439	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 5440	An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)
|- ( F : A -onto-> B -> F : A --> B )
Theorem	fofun 5441	An onto mapping is a function. (Contributed by NM, 29-Mar-2008.)
|- ( F : A -onto-> B -> Fun F )
Theorem	fofn 5442	An onto mapping is a function on its domain. (Contributed by NM, 16-Dec-2008.)
|- ( F : A -onto-> B -> F Fn A )
Theorem	forn 5443	The codomain of an onto function is its range. (Contributed by NM, 3-Aug-1994.)
|- ( F : A -onto-> B -> ran F = B )
Theorem	dffo2 5444	Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.)
|- ( F : A -onto-> B <-> ( F : A --> B /\ ran F = B ) )
Theorem	foima 5445	The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)
|- ( F : A -onto-> B -> ( F " A ) = B )
Theorem	dffn4 5446	A function maps onto its range. (Contributed by NM, 10-May-1998.)
|- ( F Fn A <-> F : A -onto-> ran F )
Theorem	funforn 5447	A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.)
|- ( Fun A <-> A : dom A -onto-> ran A )
Theorem	fodmrnu 5448	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	fores 5449	Restriction of a function. (Contributed by NM, 4-Mar-1997.)
|- ( ( Fun F /\ A C_ dom F ) -> ( F |` A ) : A -onto-> ( F " A ) )
Theorem	foco 5450	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 5451	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 5452	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 5453	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 5454	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 5455	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 5456	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 5457	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 5458	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 5459	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 5460	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 5461	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 5462	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 5463	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 5464	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 5465	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 5466	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 5467	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 5468	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 5469	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 5470	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 5471	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 ) )
Theorem	dff1o5 5472	Alternate definition of one-to-one onto function. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ ran F = B ) )
Theorem	f1orn 5473	A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.)
|- ( F : A -1-1-onto-> ran F <-> ( F Fn A /\ Fun `' F ) )
Theorem	f1f1orn 5474	A one-to-one function maps one-to-one onto its range. (Contributed by NM, 4-Sep-2004.)
|- ( F : A -1-1-> B -> F : A -1-1-onto-> ran F )
Theorem	f1oabexg 5475*	The class of all 1-1-onto functions mapping one set to another is a set. (Contributed by Paul Chapman, 25-Feb-2008.)
|- F = { f | ( f : A -1-1-onto-> B /\ ph ) }   =>    |- ( ( A e. C /\ B e. D ) -> F e. _V )
Theorem	f1ocnv 5476	The converse of a one-to-one onto function is also one-to-one onto. (Contributed by NM, 11-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|- ( F : A -1-1-onto-> B -> `' F : B -1-1-onto-> A )
Theorem	f1ocnvb 5477	A relation is a one-to-one onto function iff its converse is a one-to-one onto function with domain and codomain/range interchanged. (Contributed by NM, 8-Dec-2003.)
|- ( Rel F -> ( F : A -1-1-onto-> B <-> `' F : B -1-1-onto-> A ) )
Theorem	f1ores 5478	The restriction of a one-to-one function maps one-to-one onto the image. (Contributed by NM, 25-Mar-1998.)
|- ( ( F : A -1-1-> B /\ C C_ A ) -> ( F |` C ) : C -1-1-onto-> ( F " C ) )
Theorem	f1orescnv 5479	The converse of a one-to-one-onto restricted function. (Contributed by Paul Chapman, 21-Apr-2008.)
|- ( ( Fun `' F /\ ( F |` R ) : R -1-1-onto-> P ) -> ( `' F |` P ) : P -1-1-onto-> R )
Theorem	f1imacnv 5480	Preimage of an image. (Contributed by NM, 30-Sep-2004.)
|- ( ( F : A -1-1-> B /\ C C_ A ) -> ( `' F " ( F " C ) ) = C )
Theorem	foimacnv 5481	A reverse version of f1imacnv 5480. (Contributed by Jeff Hankins, 16-Jul-2009.)
|- ( ( F : A -onto-> B /\ C C_ B ) -> ( F " ( `' F " C ) ) = C )
Theorem	foun 5482	The union of two onto functions with disjoint domains is an onto function. (Contributed by Mario Carneiro, 22-Jun-2016.)
|- ( ( ( F : A -onto-> B /\ G : C -onto-> D ) /\ ( A i^i C ) = (/) ) -> ( F u. G ) : ( A u. C ) -onto-> ( B u. D ) )
Theorem	f1oun 5483	The union of two one-to-one onto functions with disjoint domains and ranges. (Contributed by NM, 26-Mar-1998.)
|- ( ( ( F : A -1-1-onto-> B /\ G : C -1-1-onto-> D ) /\ ( ( A i^i C ) = (/) /\ ( B i^i D ) = (/) ) ) -> ( F u. G ) : ( A u. C ) -1-1-onto-> ( B u. D ) )
Theorem	fun11iun 5484*	The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function. (Contributed by Mario Carneiro, 20-May-2013.) (Revised by Mario Carneiro, 24-Jun-2015.)
|- ( x = y -> B = C )   &    |- B e. _V   =>    |- ( A. x e. A ( B : D -1-1-> S /\ A. y e. A ( B C_ C \/ C C_ B ) ) -> U_ x e. A B : U_ x e. A D -1-1-> S )
Theorem	resdif 5485	The restriction of a one-to-one onto function to a difference maps onto the difference of the images. (Contributed by Paul Chapman, 11-Apr-2009.)
|- ( ( Fun `' F /\ ( F |` A ) : A -onto-> C /\ ( F |` B ) : B -onto-> D ) -> ( F |` ( A \ B ) ) : ( A \ B ) -1-1-onto-> ( C \ D ) )
Theorem	f1oco 5486	Composition of one-to-one onto functions. (Contributed by NM, 19-Mar-1998.)
|- ( ( F : B -1-1-onto-> C /\ G : A -1-1-onto-> B ) -> ( F o. G ) : A -1-1-onto-> C )
Theorem	f1cnv 5487	The converse of an injective function is bijective. (Contributed by FL, 11-Nov-2011.)
|- ( F : A -1-1-> B -> `' F : ran F -1-1-onto-> A )
Theorem	funcocnv2 5488	Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( Fun F -> ( F o. `' F ) = ( _I |` ran F ) )
Theorem	fococnv2 5489	The composition of an onto function and its converse. (Contributed by Stefan O'Rear, 12-Feb-2015.)
|- ( F : A -onto-> B -> ( F o. `' F ) = ( _I |` B ) )
Theorem	f1ococnv2 5490	The composition of a one-to-one onto function and its converse equals the identity relation restricted to the function's range. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Stefan O'Rear, 12-Feb-2015.)
|- ( F : A -1-1-onto-> B -> ( F o. `' F ) = ( _I |` B ) )
Theorem	f1cocnv2 5491	Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
|- ( F : A -1-1-> B -> ( F o. `' F ) = ( _I |` ran F ) )
Theorem	f1ococnv1 5492	The composition of a one-to-one onto function's converse and itself equals the identity relation restricted to the function's domain. (Contributed by NM, 13-Dec-2003.)
|- ( F : A -1-1-onto-> B -> ( `' F o. F ) = ( _I |` A ) )
Theorem	f1cocnv1 5493	Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
|- ( F : A -1-1-> B -> ( `' F o. F ) = ( _I |` A ) )
Theorem	funcoeqres 5494	Express a constraint on a composition as a constraint on the composand. (Contributed by Stefan O'Rear, 7-Mar-2015.)
|- ( ( Fun G /\ ( F o. G ) = H ) -> ( F |` ran G ) = ( H o. `' G ) )
Theorem	ffoss 5495*	Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145. (Contributed by NM, 10-May-1998.)
|- F e. _V   =>    |- ( F : A --> B <-> E. x ( F : A -onto-> x /\ x C_ B ) )
Theorem	f11o 5496*	Relationship between one-to-one and one-to-one onto function. (Contributed by NM, 4-Apr-1998.)
|- F e. _V   =>    |- ( F : A -1-1-> B <-> E. x ( F : A -1-1-onto-> x /\ x C_ B ) )
Theorem	f10 5497	The empty set maps one-to-one into any class. (Contributed by NM, 7-Apr-1998.)
|- (/) : (/) -1-1-> A
Theorem	f1o00 5498	One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.)
|- ( F : (/) -1-1-onto-> A <-> ( F = (/) /\ A = (/) ) )
Theorem	fo00 5499	Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.)
|- ( F : (/) -onto-> A <-> ( F = (/) /\ A = (/) ) )
Theorem	f1o0 5500	One-to-one onto mapping of the empty set. (Contributed by NM, 10-Sep-2004.)
|- (/) : (/) -1-1-onto-> (/)