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Theorem List for Metamath Proof Explorer - 5301-5400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfco 5301 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 )
 
Theoremfco2 5302 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 )
 
Theoremfssxp 5303 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 ) )
 
Theoremfex2 5304 A function with bounded domain and range is a set. This version of fex 5648 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 )
 
Theoremfunssxp 5305 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 ) )
 
Theoremffdm 5306 A mapping is a partial function. (Contributed by NM, 25-Nov-2007.)
 |-  ( F : A --> B  ->  ( F : dom  F --> B  /\  dom  F 
 C_  A ) )
 
Theoremopelf 5307 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 ) )
 
Theoremfun 5308 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 ) )
 
Theoremfun2 5309 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 )
 
Theoremfnfco 5310 Composition of two functions. (Contributed by NM, 22-May-2006.)
 |-  ( ( F  Fn  A  /\  G : B --> A )  ->  ( F  o.  G )  Fn  B )
 
Theoremfssres 5311 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 )
 
Theoremfssres2 5312 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 )
 
Theoremfresin 5313 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 )
 
Theoremresasplit 5314 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 ) ) ) ) )
 
Theoremfresaun 5315 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 )
 
Theoremfresaunres2 5316 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 )
 
Theoremfresaunres1 5317 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 )
 
Theoremfcoi1 5318 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 )
 
Theoremfcoi2 5319 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 )
 
Theoremfeu 5320* 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 )
 
Theoremfcnvres 5321 The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.)
 |-  ( F : A --> B  ->  `' ( F  |`  A )  =  ( `' F  |`  B ) )
 
Theoremfimacnvdisj 5322 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 )  =  (/) )
 
Theoremfint 5323* 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 )
 
Theoremfin 5324 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 ) )
 
Theoremfabexg 5325* 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 )
 
Theoremfabex 5326* 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
 
Theoremdmfex 5327 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 )
 
Theoremf0 5328 The empty function. (Contributed by NM, 14-Aug-1999.)
 |-  (/) : (/) --> A
 
Theoremf00 5329 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  =  (/) ) )
 
Theoremfconst 5330 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 }
 
Theoremfconstg 5331 A cross product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.)
 |-  ( B  e.  V  ->  ( A  X.  { B } ) : A --> { B } )
 
Theoremfnconstg 5332 A cross product with a singleton is a constant function. (Contributed by NM, 24-Jul-2014.)
 |-  ( B  e.  V  ->  ( A  X.  { B } )  Fn  A )
 
Theoremfconst6g 5333 Constant function with loose range. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( B  e.  C  ->  ( A  X.  { B } ) : A --> C )
 
Theoremfconst6 5334 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
 
Theoremf1eq1 5335 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 ) )
 
Theoremf1eq2 5336 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 ) )
 
Theoremf1eq3 5337 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 ) )
 
Theoremnff1 5338 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
 
Theoremdff12 5339* 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 ) )
 
Theoremf1f 5340 A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)
 |-  ( F : A -1-1-> B 
 ->  F : A --> B )
 
Theoremf1fn 5341 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 )
 
Theoremf1fun 5342 A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)
 |-  ( F : A -1-1-> B 
 ->  Fun  F )
 
Theoremf1rel 5343 A one-to-one onto mapping is a relation. (Contributed by NM, 8-Mar-2014.)
 |-  ( F : A -1-1-> B 
 ->  Rel  F )
 
Theoremf1dm 5344 The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.)
 |-  ( F : A -1-1-> B 
 ->  dom  F  =  A )
 
Theoremf1ss 5345 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 )
 
Theoremf1ssr 5346 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 )
 
Theoremf1ssres 5347 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 )
 
Theoremf1cnvcnv 5348 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 ) )
 
Theoremf1co 5349 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 )
 
Theoremfoeq1 5350 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
 |-  ( F  =  G  ->  ( F : A -onto-> B 
 <->  G : A -onto-> B ) )
 
Theoremfoeq2 5351 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  =  B  ->  ( F : A -onto-> C 
 <->  F : B -onto-> C ) )
 
Theoremfoeq3 5352 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  =  B  ->  ( F : C -onto-> A 
 <->  F : C -onto-> B ) )
 
Theoremnffo 5353 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
 
Theoremfof 5354 An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)
 |-  ( F : A -onto-> B  ->  F : A --> B )
 
Theoremfofun 5355 An onto mapping is a function. (Contributed by NM, 29-Mar-2008.)
 |-  ( F : A -onto-> B  ->  Fun  F )
 
Theoremfofn 5356 An onto mapping is a function on its domain. (Contributed by NM, 16-Dec-2008.)
 |-  ( F : A -onto-> B  ->  F  Fn  A )
 
Theoremforn 5357 The codomain of an onto function is its range. (Contributed by NM, 3-Aug-1994.)
 |-  ( F : A -onto-> B  ->  ran  F  =  B )
 
Theoremdffo2 5358 Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.)
 |-  ( F : A -onto-> B 
 <->  ( F : A --> B  /\  ran  F  =  B ) )
 
Theoremfoima 5359 The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)
 |-  ( F : A -onto-> B  ->  ( F " A )  =  B )
 
Theoremdffn4 5360 A function maps onto its range. (Contributed by NM, 10-May-1998.)
 |-  ( F  Fn  A  <->  F : A -onto-> ran  F )
 
Theoremfunforn 5361 A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.)
 |-  ( Fun  A  <->  A : dom  A -onto-> ran  A )
 
Theoremfodmrnu 5362 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 )
 )
 
Theoremfores 5363 Restriction of a function. (Contributed by NM, 4-Mar-1997.)
 |-  ( ( Fun  F  /\  A  C_  dom  F ) 
 ->  ( F  |`  A ) : A -onto-> ( F
 " A ) )
 
Theoremfoco 5364 Composition of onto functions. (Contributed by NM, 22-Mar-2006.)
 |-  ( ( F : B -onto-> C  /\  G : A -onto-> B )  ->  ( F  o.  G ) : A -onto-> C )
 
Theoremfoconst 5365 A nonzero constant function is onto. (Contributed by NM, 12-Jan-2007.)
 |-  ( ( F : A
 --> { B }  /\  F  =/=  (/) )  ->  F : A -onto-> { B } )
 
Theoremf1oeq1 5366 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 ) )
 
Theoremf1oeq2 5367 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 ) )
 
Theoremf1oeq3 5368 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 ) )
 
Theoremf1oeq23 5369 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 ) )
 
Theoremf1eq123d 5370 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 ) )
 
Theoremfoeq123d 5371 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 ) )
 
Theoremf1oeq123d 5372 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 ) )
 
Theoremnff1o 5373 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
 
Theoremf1of1 5374 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 )
 
Theoremf1of 5375 A one-to-one onto mapping is a mapping. (Contributed by NM, 12-Dec-2003.)
 |-  ( F : A -1-1-onto-> B  ->  F : A --> B )
 
Theoremf1ofn 5376 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 )
 
Theoremf1ofun 5377 A one-to-one onto mapping is a function. (Contributed by NM, 12-Dec-2003.)
 |-  ( F : A -1-1-onto-> B  ->  Fun  F )
 
Theoremf1orel 5378 A one-to-one onto mapping is a relation. (Contributed by NM, 13-Dec-2003.)
 |-  ( F : A -1-1-onto-> B  ->  Rel  F )
 
Theoremf1odm 5379 The domain of a one-to-one onto mapping. (Contributed by NM, 8-Mar-2014.)
 |-  ( F : A -1-1-onto-> B  ->  dom  F  =  A )
 
Theoremdff1o2 5380 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 ) )
 
Theoremdff1o3 5381 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 ) )
 
Theoremf1ofo 5382 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 )
 
Theoremdff1o4 5383 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 ) )
 
Theoremdff1o5 5384 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 ) )
 
Theoremf1orn 5385 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 ) )
 
Theoremf1f1orn 5386 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 )
 
Theoremf1oabexg 5387* 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 )
 
Theoremf1ocnv 5388 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 )
 
Theoremf1ocnvb 5389 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 ) )
 
Theoremf1ores 5390 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 ) )
 
Theoremf1orescnv 5391 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 )
 
Theoremf1imacnv 5392 Preimage of an image. (Contributed by NM, 30-Sep-2004.)
 |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( `' F " ( F
 " C ) )  =  C )
 
Theoremfoimacnv 5393 A reverse version of f1imacnv 5392. (Contributed by Jeffrey Hankins, 16-Jul-2009.)
 |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  ( F
 " ( `' F " C ) )  =  C )
 
Theoremfoun 5394 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 ) )
 
Theoremf1oun 5395 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 ) )
 
Theoremfun11iun 5396* 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 )
 
Theoremresdif 5397 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 ) )
 
Theoremresin 5398 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 ) )
 
Theoremf1oco 5399 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 )
 
Theoremf1cnv 5400 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 )
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