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	fssxp 5001	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 5002	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 5003	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 5004	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 5005	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 5006	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 5007	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 5008	Composition of two functions. (Contributed by NM, 22-May-2006.)
|- (( F Fn A /\ G :B -->A) -> ( F o. G) Fn B)
Theorem	fssres 5009	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 5010	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 5011	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 5012	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 5013	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 5014	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 5015*	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 5016	The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.)
|- ( F :A -->B -> `'( F |` A) = ( `' F |` B))
Theorem	fimacnvdisj 5017	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 5018*	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 5019	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 5020*	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 5021*	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 5022	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 5023	The empty function. (Contributed by NM, 14-Aug-1999.)
|- (/) : (/) -->A
Theorem	f00 5024	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 5025	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 5026	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 5027	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 5028	Constant function with loose range. (Contributed by Stefan O'Rear, 1-Feb-2015.)
|- (B e. C -> (A X. {B }) :A --> C)
Theorem	fconst6 5029	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 5030	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 5031	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 5032	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 5033	Bound-variable hypothesis builder for a one-to-one function. (Contributed by NM, 16-May-2004.)
|- F/_x F   &   |- F/_xA   &   |- F/_xB   =>   |- F/x F :A -1-1->B
Theorem	dff12 5034*	Alternate definition of a one-to-one function. (Contributed by NM, 31-Dec-1996.)
|- ( F :A -1-1->B <-> ( F :A -->B /\ A.yE*x x Fy))
Theorem	f1f 5035	A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)
|- ( F :A -1-1->B -> F :A -->B)
Theorem	f1fn 5036	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 5037	A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)
|- ( F :A -1-1->B -> Fun F)
Theorem	f1rel 5038	A one-to-one onto mapping is a relation. (Contributed by NM, 8-Mar-2014.)
|- ( F :A -1-1->B -> Rel F)
Theorem	f1dm 5039	The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.)
|- ( F :A -1-1->B -> dom F = A)
Theorem	f1ss 5040	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 5041	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 5042	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 5043	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 5044	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 5045	Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
|- ( F = G -> ( F :A -onto->B <-> G :A -onto->B))
Theorem	foeq2 5046	Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
|- (A = B -> ( F :A -onto-> C <-> F :B -onto-> C))
Theorem	foeq3 5047	Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
|- (A = B -> ( F : C -onto->A <-> F : C -onto->B))
Theorem	nffo 5048	Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.)
|- F/_x F   &   |- F/_xA   &   |- F/_xB   =>   |- F/x F :A -onto->B
Theorem	fof 5049	An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)
|- ( F :A -onto->B -> F :A -->B)
Theorem	fofun 5050	An onto mapping is a function. (Contributed by NM, 29-Mar-2008.)
|- ( F :A -onto->B -> Fun F)
Theorem	fofn 5051	An onto mapping is a function on its domain. (Contributed by NM, 16-Dec-2008.)
|- ( F :A -onto->B -> F Fn A)
Theorem	forn 5052	The codomain of an onto function is its range. (Contributed by NM, 3-Aug-1994.)
|- ( F :A -onto->B -> ran F = B)
Theorem	dffo2 5053	Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.)
|- ( F :A -onto->B <-> ( F :A -->B /\ ran F = B))
Theorem	foima 5054	The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)
|- ( F :A -onto->B -> ( F "A) = B)
Theorem	dffn4 5055	A function maps onto its range. (Contributed by NM, 10-May-1998.)
|- ( F Fn A <-> F :A -onto-> ran F)
Theorem	funforn 5056	A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.)
|- ( Fun A <-> A : dom A -onto-> ran A)
Theorem	fodmrnu 5057	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 5058	Restriction of a function. (Contributed by NM, 4-Mar-1997.)
|- (( Fun F /\ A C_ dom F) -> ( F |` A) :A -onto->( F "A))
Theorem	foco 5059	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 5060	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 5061	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 5062	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 5063	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 5064	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 5065	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 5066	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	nff1o 5067	Bound-variable hypothesis builder for a one-to-one onto function. (Contributed by NM, 16-May-2004.)
|- F/_x F   &   |- F/_xA   &   |- F/_xB   =>   |- F/x F :A -1-1-onto->B
Theorem	f1of1 5068	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 5069	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 5070	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 5071	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 5072	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 5073	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 5074	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 5075	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 5076	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 5077	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 5078	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 5079	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 5080	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 5081*	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 5082	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 5083	A relation is a one-to-one onto function iff its converse is a one-to-one onto function with domain and range interchanged. (Contributed by NM, 8-Dec-2003.)
|- ( Rel F -> ( F :A -1-1-onto->B <-> `' F :B -1-1-onto->A))
Theorem	f1ores 5084	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 5085	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 5086	Preimage of an image. (Contributed by NM, 30-Sep-2004.)
|- (( F :A -1-1->B /\ C C_ A) -> ( `' F "( F " C)) = C)
Theorem	foimacnv 5087	A reverse version of f1imacnv 5086. (Contributed by Jeff Hankins, 16-Jul-2009.)
|- (( F :A -onto->B /\ C C_ B) -> ( F "( `' F " C)) = C)
Theorem	foun 5088	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 5089	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 5090*	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 5091	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 5092	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 5093	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 5094	Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( Fun F -> ( F o. `' F) = ( _I |` ran F))
Theorem	fococnv2 5095	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 5096	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 5097	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 5098	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 5099	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 5100	Re-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))