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Theorem List for Metamath Proof Explorer - 5301-5400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfneq1d 5301 Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  ->  F  =  G )   =>    |-  ( ph  ->  ( F  Fn  A  <->  G  Fn  A ) )
 
Theoremfneq2d 5302 Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F  Fn  A  <->  F  Fn  B ) )
 
Theoremfneq12d 5303 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 ) )
 
Theoremfneq1i 5304 Equality inference for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  F  =  G   =>    |-  ( F  Fn  A 
 <->  G  Fn  A )
 
Theoremfneq2i 5305 Equality inference for function predicate with domain. (Contributed by NM, 4-Sep-2011.)
 |-  A  =  B   =>    |-  ( F  Fn  A 
 <->  F  Fn  B )
 
Theoremnffn 5306 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
 
Theoremfnfun 5307 A function with domain is a function. (Contributed by NM, 1-Aug-1994.)
 |-  ( F  Fn  A  ->  Fun  F )
 
Theoremfnrel 5308 A function with domain is a relation. (Contributed by NM, 1-Aug-1994.)
 |-  ( F  Fn  A  ->  Rel  F )
 
Theoremfndm 5309 The domain of a function. (Contributed by NM, 2-Aug-1994.)
 |-  ( F  Fn  A  ->  dom  F  =  A )
 
Theoremfunfni 5310 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 )
 
Theoremfndmu 5311 A function has a unique domain. (Contributed by NM, 11-Aug-1994.)
 |-  ( ( F  Fn  A  /\  F  Fn  B )  ->  A  =  B )
 
Theoremfnbr 5312 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 )
 
Theoremfnop 5313 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 )
 
Theoremfneu 5314* 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 )
 
Theoremfneu2 5315* 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 )
 
Theoremfnun 5316 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 ) )
 
Theoremfnunsn 5317 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 )
 
Theoremfnco 5318 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 )
 
Theoremfnresdm 5319 A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.)
 |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
 
Theoremfnresdisj 5320 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 )  =  (/) ) )
 
Theorem2elresin 5321 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 ) ) ) ) )
 
Theoremfnssresb 5322 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 ) )
 
Theoremfnssres 5323 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 )
 
Theoremfnresin1 5324 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 ) )
 
Theoremfnresin2 5325 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 ) )
 
Theoremfnres 5326* An equivalence for functionality of a restriction. Compare dffun8 5247. (Contributed by Mario Carneiro, 20-May-2015.)
 |-  ( ( F  |`  A )  Fn  A  <->  A. x  e.  A  E! y  x F y )
 
Theoremfnresi 5327 Functionality and domain of restricted identity. (Contributed by NM, 27-Aug-2004.)
 |-  (  _I  |`  A )  Fn  A
 
Theoremfnima 5328 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 )
 
Theoremfn0 5329 A function with empty domain is empty. (Contributed by NM, 15-Apr-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 |-  ( F  Fn  (/)  <->  F  =  (/) )
 
Theoremfnimadisj 5330 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 )  =  (/) )
 
Theoremfnimaeq0 5331 Images under a function never map nonempty sets to empty sets. EDITORIAL: usable in fnwe2lem2 26559. (Contributed by Stefan O'Rear, 21-Jan-2015.)
 |-  ( ( F  Fn  A  /\  B  C_  A )  ->  ( ( F
 " B )  =  (/) 
 <->  B  =  (/) ) )
 
Theoremdfmpt3 5332 Alternate definition for the "maps to" notation df-mpt 4080. (Contributed by Mario Carneiro, 30-Dec-2016.)
 |-  ( x  e.  A  |->  B )  =  U_ x  e.  A  ( { x }  X.  { B }
 )
 
Theoremfnopabg 5333* 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 )
 
Theoremfnopab 5334* 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
 
Theoremmptfng 5335* 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 )
 
Theoremfnmpt 5336* 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 )
 
Theoremmpt0 5337 A mapping operation with empty domain. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  ( x  e.  (/)  |->  A )  =  (/)
 
Theoremfnmpti 5338* 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
 
Theoremdmmpti 5339* 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
 
Theoremmptun 5340 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 ) )
 
Theoremfeq1 5341 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
 |-  ( F  =  G  ->  ( F : A --> B 
 <->  G : A --> B ) )
 
Theoremfeq2 5342 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  =  B  ->  ( F : A --> C 
 <->  F : B --> C ) )
 
Theoremfeq3 5343 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  =  B  ->  ( F : C --> A 
 <->  F : C --> B ) )
 
Theoremfeq23 5344 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 ) )
 
Theoremfeq1d 5345 Equality deduction for functions. (Contributed by NM, 19-Feb-2008.)
 |-  ( ph  ->  F  =  G )   =>    |-  ( ph  ->  ( F : A --> B  <->  G : A --> B ) )
 
Theoremfeq2d 5346 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F : A --> C  <->  F : B --> C ) )
 
Theoremfeq12d 5347 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F : A --> C  <->  G : B --> C ) )
 
Theoremfeq123d 5348 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 ) )
 
Theoremfeq1i 5349 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  F  =  G   =>    |-  ( F : A
 --> B  <->  G : A --> B )
 
Theoremfeq2i 5350 Equality inference for functions. (Contributed by NM, 5-Sep-2011.)
 |-  A  =  B   =>    |-  ( F : A
 --> C  <->  F : B --> C )
 
Theoremfeq23i 5351 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  A  =  C   &    |-  B  =  D   =>    |-  ( F : A --> B 
 <->  F : C --> D )
 
Theoremfeq23d 5352 Equality deduction for functions. (Contributed by NM, 8-Jun-2013.)
 |-  ( ph  ->  A  =  C )   &    |-  ( ph  ->  B  =  D )   =>    |-  ( ph  ->  ( F : A --> B  <->  F : C --> D ) )
 
Theoremnff 5353 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
 
Theoremelimf 5354 Eliminate a mapping hypothesis for the weak deduction theorem dedth 3607, when a special case  G : A --> B is provable, in order to convert  F : A --> B from a hypothesis to an antecedent. (Contributed by NM, 24-Aug-2006.)
 |-  G : A --> B   =>    |-  if ( F : A --> B ,  F ,  G ) : A --> B
 
Theoremffn 5355 A mapping is a function. (Contributed by NM, 2-Aug-1994.)
 |-  ( F : A --> B  ->  F  Fn  A )
 
Theoremdffn2 5356 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 )
 
Theoremffun 5357 A mapping is a function. (Contributed by NM, 3-Aug-1994.)
 |-  ( F : A --> B  ->  Fun  F )
 
Theoremfrel 5358 A mapping is a relation. (Contributed by NM, 3-Aug-1994.)
 |-  ( F : A --> B  ->  Rel  F )
 
Theoremfdm 5359 The domain of a mapping. (Contributed by NM, 2-Aug-1994.)
 |-  ( F : A --> B  ->  dom  F  =  A )
 
Theoremfdmi 5360 The domain of a mapping. (Contributed by NM, 28-Jul-2008.)
 |-  F : A --> B   =>    |-  dom  F  =  A
 
Theoremfrn 5361 The range of a mapping. (Contributed by NM, 3-Aug-1994.)
 |-  ( F : A --> B  ->  ran  F  C_  B )
 
Theoremdffn3 5362 A function maps to its range. (Contributed by NM, 1-Sep-1999.)
 |-  ( F  Fn  A  <->  F : A --> ran  F )
 
Theoremfss 5363 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 )
 
Theoremfco 5364 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 5365 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 5366 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 5367 A function with bounded domain and range is a set. This version of fex 5711 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 5368 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 5369 A mapping is a partial function. (Contributed by NM, 25-Nov-2007.)
 |-  ( F : A --> B  ->  ( F :  dom  F --> B  /\  dom  F 
 C_  A ) )
 
Theoremopelf 5370 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 5371 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 5372 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 5373 Composition of two functions. (Contributed by NM, 22-May-2006.)
 |-  ( ( F  Fn  A  /\  G : B --> A )  ->  ( F  o.  G )  Fn  B )
 
Theoremfssres 5374 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 5375 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 5376 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 5377 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 5378 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 5379 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 5380 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 5381 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 5382 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 5383* 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 5384 The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.)
 |-  ( F : A --> B  ->  `' ( F  |`  A )  =  ( `' F  |`  B ) )
 
Theoremfimacnvdisj 5385 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 5386* 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 5387 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 5388* 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 5389* 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 5390 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 5391 The empty function. (Contributed by NM, 14-Aug-1999.)
 |-  (/) : (/) --> A
 
Theoremf00 5392 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 5393 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 5394 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 5395 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 5396 Constant function with loose range. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( B  e.  C  ->  ( A  X.  { B } ) : A --> C )
 
Theoremfconst6 5397 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 5398 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 5399 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 5400 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 ) )
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