P. 62 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 6101-6200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	f1f 6101	A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)
|- ( F : A -1-1-> B -> F : A --> B )
Theorem	f1fn 6102	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 6103	A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)
|- ( F : A -1-1-> B -> Fun F )
Theorem	f1rel 6104	A one-to-one onto mapping is a relation. (Contributed by NM, 8-Mar-2014.)
|- ( F : A -1-1-> B -> Rel F )
Theorem	f1dm 6105	The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.)
|- ( F : A -1-1-> B -> dom F = A )
Theorem	f1ss 6106	A function that is one-to-one is also one-to-one on some superset of its codomain. (Contributed by Mario Carneiro, 12-Jan-2013.)
|- ( ( F : A -1-1-> B /\ B C_ C ) -> F : A -1-1-> C )
Theorem	f1ssr 6107	A function that is one-to-one is also one-to-one on some superset of its range. (Contributed by Stefan O'Rear, 20-Feb-2015.)
|- ( ( F : A -1-1-> B /\ ran F C_ C ) -> F : A -1-1-> C )
Theorem	f1ssres 6108	A function that is one-to-one is also one-to-one on some subset 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 6109	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 6110	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 6111	Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
|- ( F = G -> ( F : A -onto-> B <-> G : A -onto-> B ) )
Theorem	foeq2 6112	Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
|- ( A = B -> ( F : A -onto-> C <-> F : B -onto-> C ) )
Theorem	foeq3 6113	Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
|- ( A = B -> ( F : C -onto-> A <-> F : C -onto-> B ) )
Theorem	nffo 6114	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 6115	An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)
|- ( F : A -onto-> B -> F : A --> B )
Theorem	fofun 6116	An onto mapping is a function. (Contributed by NM, 29-Mar-2008.)
|- ( F : A -onto-> B -> Fun F )
Theorem	fofn 6117	An onto mapping is a function on its domain. (Contributed by NM, 16-Dec-2008.)
|- ( F : A -onto-> B -> F Fn A )
Theorem	forn 6118	The codomain of an onto function is its range. (Contributed by NM, 3-Aug-1994.)
|- ( F : A -onto-> B -> ran F = B )
Theorem	dffo2 6119	Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.)
|- ( F : A -onto-> B <-> ( F : A --> B /\ ran F = B ) )
Theorem	foima 6120	The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)
|- ( F : A -onto-> B -> ( F " A ) = B )
Theorem	dffn4 6121	A function maps onto its range. (Contributed by NM, 10-May-1998.)
|- ( F Fn A <-> F : A -onto-> ran F )
Theorem	funforn 6122	A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.)
|- ( Fun A <-> A : dom A -onto-> ran A )
Theorem	fodmrnu 6123	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 6124	Restriction of an onto function. (Contributed by NM, 4-Mar-1997.)
|- ( ( Fun F /\ A C_ dom F ) -> ( F |` A ) : A -onto-> ( F " A ) )
Theorem	foco 6125	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	foconst 6126	A nonzero constant function is onto. (Contributed by NM, 12-Jan-2007.)
|- ( ( F : A --> { B } /\ F =/= (/) ) -> F : A -onto-> { B } )
Theorem	f1oeq1 6127	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 6128	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 6129	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 6130	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 6131	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 6132	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 6133	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	f1oeq3d 6134	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 6135	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 6136	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 6137	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 6138	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 6139	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 6140	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 6141	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 6142	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 6143	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 6144	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 6145	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 6146	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 6147	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 6148	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	f1ocnv 6149	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 6150	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 6151	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 6152	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 6153	Preimage of an image. (Contributed by NM, 30-Sep-2004.)
|- ( ( F : A -1-1-> B /\ C C_ A ) -> ( `' F " ( F " C ) ) = C )
Theorem	foimacnv 6154	A reverse version of f1imacnv 6153. (Contributed by Jeff Hankins, 16-Jul-2009.)
|- ( ( F : A -onto-> B /\ C C_ B ) -> ( F " ( `' F " C ) ) = C )
Theorem	foun 6155	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 6156	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	resdif 6157	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	resin 6158	The restriction of a one-to-one onto function to an intersection maps onto the intersection of the images. (Contributed by Paul Chapman, 11-Apr-2009.)
|- ( ( Fun `' F /\ ( F |` A ) : A -onto-> C /\ ( F |` B ) : B -onto-> D ) -> ( F |` ( A i^i B ) ) : ( A i^i B ) -1-1-onto-> ( C i^i D ) )
Theorem	f1oco 6159	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 6160	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 6161	Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( Fun F -> ( F o. `' F ) = ( _I |` ran F ) )
Theorem	fococnv2 6162	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 6163	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 6164	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 6165	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 6166	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 6167	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 ) )
Theorem	f1ssf1 6168	A subset of an injective function is injective. (Contributed by AV, 20-Nov-2020.)
|- ( ( Fun F /\ Fun `' F /\ G C_ F ) -> Fun `' G )
Theorem	f10 6169	The empty set maps one-to-one into any class. (Contributed by NM, 7-Apr-1998.)
|- (/) : (/) -1-1-> A
Theorem	f10d 6170	The empty set maps one-to-one into any class, deduction version. (Contributed by AV, 25-Nov-2020.)
|- ( ph -> F = (/) )   =>    |- ( ph -> F : dom F -1-1-> A )
Theorem	f1o00 6171	One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.)
|- ( F : (/) -1-1-onto-> A <-> ( F = (/) /\ A = (/) ) )
Theorem	fo00 6172	Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.)
|- ( F : (/) -onto-> A <-> ( F = (/) /\ A = (/) ) )
Theorem	f1o0 6173	One-to-one onto mapping of the empty set. (Contributed by NM, 10-Sep-2004.)
|- (/) : (/) -1-1-onto-> (/)
Theorem	f1oi 6174	A restriction of the identity relation is a one-to-one onto function. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|- ( _I |` A ) : A -1-1-onto-> A
Theorem	f1ovi 6175	The identity relation is a one-to-one onto function on the universe. (Contributed by NM, 16-May-2004.)
|- _I : _V -1-1-onto-> _V
Theorem	f1osn 6176	A singleton of an ordered pair is one-to-one onto function. (Contributed by NM, 18-May-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|- A e. _V   &    |- B e. _V   =>    |- { <. A , B >. } : { A } -1-1-onto-> { B }
Theorem	f1osng 6177	A singleton of an ordered pair is one-to-one onto function. (Contributed by Mario Carneiro, 12-Jan-2013.)
|- ( ( A e. V /\ B e. W ) -> { <. A , B >. } : { A } -1-1-onto-> { B } )
Theorem	f1sng 6178	A singleton of an ordered pair is a one-to-one function. (Contributed by AV, 17-Apr-2021.)
|- ( ( A e. V /\ B e. W ) -> { <. A , B >. } : { A } -1-1-> W )
Theorem	fsnd 6179	A singleton of an ordered pair is a function. (Contributed by AV, 17-Apr-2021.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   =>    |- ( ph -> { <. A , B >. } : { A } --> W )
Theorem	f1oprswap 6180	A two-element swap is a bijection on a pair. (Contributed by Mario Carneiro, 23-Jan-2015.)
|- ( ( A e. V /\ B e. W ) -> { <. A , B >. , <. B , A >. } : { A , B } -1-1-onto-> { A , B } )
Theorem	f1oprg 6181	An unordered pair of ordered pairs with different elements is a one-to-one onto function, analogous to f1oprswap 6180. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
|- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) -> ( ( A =/= C /\ B =/= D ) -> { <. A , B >. , <. C , D >. } : { A , C } -1-1-onto-> { B , D } ) )
Theorem	tz6.12-2 6182*	Function value when F is not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
|- ( -. E! x A F x -> ( F ` A ) = (/) )
Theorem	fveu 6183*	The value of a function at a unique point. (Contributed by Scott Fenton, 6-Oct-2017.)
|- ( E! x A F x -> ( F ` A ) = U. { x | A F x } )
Theorem	brprcneu 6184*	If A is a proper class and F is any class, then there is no unique set which is related to A through the binary relation F. (Contributed by Scott Fenton, 7-Oct-2017.)
|- ( -. A e. _V -> -. E! x A F x )
Theorem	fvprc 6185	A function's value at a proper class is the empty set. (Contributed by NM, 20-May-1998.)
|- ( -. A e. _V -> ( F ` A ) = (/) )
Theorem	fv2 6186*	Alternate definition of function value. Definition 10.11 of [Quine] p. 68. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- ( F ` A ) = U. { x | A. y ( A F y <-> y = x ) }
Theorem	dffv3 6187*	A definition of function value in terms of iota. (Contributed by Scott Fenton, 19-Feb-2013.)
|- ( F ` A ) = ( iota x x e. ( F " { A } ) )
Theorem	dffv4 6188*	The previous definition of function value, from before the iota operator was introduced. Although based on the idea embodied by Definition 10.2 of [Quine] p. 65 (see args 5493), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.)
|- ( F ` A ) = U. { x | ( F " { A } ) = { x } }
Theorem	elfv 6189*	Membership in a function value. (Contributed by NM, 30-Apr-2004.)
|- ( A e. ( F ` B ) <-> E. x ( A e. x /\ A. y ( B F y <-> y = x ) ) )
Theorem	fveq1 6190	Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
|- ( F = G -> ( F ` A ) = ( G ` A ) )
Theorem	fveq2 6191	Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
|- ( A = B -> ( F ` A ) = ( F ` B ) )
Theorem	fveq1i 6192	Equality inference for function value. (Contributed by NM, 2-Sep-2003.)
|- F = G   =>    |- ( F ` A ) = ( G ` A )
Theorem	fveq1d 6193	Equality deduction for function value. (Contributed by NM, 2-Sep-2003.)
|- ( ph -> F = G )   =>    |- ( ph -> ( F ` A ) = ( G ` A ) )
Theorem	fveq2i 6194	Equality inference for function value. (Contributed by NM, 28-Jul-1999.)
|- A = B   =>    |- ( F ` A ) = ( F ` B )
Theorem	fveq2d 6195	Equality deduction for function value. (Contributed by NM, 29-May-1999.)
|- ( ph -> A = B )   =>    |- ( ph -> ( F ` A ) = ( F ` B ) )
Theorem	fveq12i 6196	Equality deduction for function value. (Contributed by FL, 27-Jun-2014.)
|- F = G   &    |- A = B   =>    |- ( F ` A ) = ( G ` B )
Theorem	fveq12d 6197	Equality deduction for function value. (Contributed by FL, 22-Dec-2008.)
|- ( ph -> F = G )   &    |- ( ph -> A = B )   =>    |- ( ph -> ( F ` A ) = ( G ` B ) )
Theorem	nffv 6198	Bound-variable hypothesis builder for function value. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- F/_ x F   &    |- F/_ x A   =>    |- F/_ x ( F ` A )
Theorem	nffvmpt1 6199*	Bound-variable hypothesis builder for mapping, special case. (Contributed by Mario Carneiro, 25-Dec-2016.)
|- F/_ x ( ( x e. A |-> B ) ` C )
Theorem	nffvd 6200	Deduction version of bound-variable hypothesis builder nffv 6198. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- ( ph -> F/_ x F )   &    |- ( ph -> F/_ x A )   =>    |- ( ph -> F/_ x ( F ` A ) )