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Theorem List for Intuitionistic Logic Explorer - 5301-5400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremffvelrni 5301 A function's value belongs to its codomain. (Contributed by NM, 6-Apr-2005.)
 |-  F : A --> B   =>    |-  ( C  e.  A  ->  ( F `  C )  e.  B )
 
Theoremffvelrnda 5302 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
 |-  ( ph  ->  F : A --> B )   =>    |-  ( ( ph  /\  C  e.  A ) 
 ->  ( F `  C )  e.  B )
 
Theoremffvelrnd 5303 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
 |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  C  e.  A )   =>    |-  ( ph  ->  ( F `  C )  e.  B )
 
Theoremrexrn 5304* Restricted existential quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)
 |-  ( x  =  ( F `  y ) 
 ->  ( ph  <->  ps ) )   =>    |-  ( F  Fn  A  ->  ( E. x  e.  ran  F ph  <->  E. y  e.  A  ps ) )
 
Theoremralrn 5305* Restricted universal quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)
 |-  ( x  =  ( F `  y ) 
 ->  ( ph  <->  ps ) )   =>    |-  ( F  Fn  A  ->  ( A. x  e.  ran  F ph  <->  A. y  e.  A  ps ) )
 
Theoremelrnrexdm 5306* For any element in the range of a function there is an element in the domain of the function for which the function value is the element of the range. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
 |-  ( Fun  F  ->  ( Y  e.  ran  F  ->  E. x  e.  dom  F  Y  =  ( F `
  x ) ) )
 
Theoremelrnrexdmb 5307* For any element in the range of a function there is an element in the domain of the function for which the function value is the element of the range. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
 |-  ( Fun  F  ->  ( Y  e.  ran  F  <->  E. x  e.  dom  F  Y  =  ( F `  x ) ) )
 
Theoremeldmrexrn 5308* For any element in the domain of a function there is an element in the range of the function which is the function value for the element of the domain. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
 |-  ( Fun  F  ->  ( Y  e.  dom  F  ->  E. x  e.  ran  F  x  =  ( F `
  Y ) ) )
 
Theoremralrnmpt 5309* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  F  =  ( x  e.  A  |->  B )   &    |-  ( y  =  B  ->  ( ps  <->  ch ) )   =>    |-  ( A. x  e.  A  B  e.  V  ->  ( A. y  e. 
 ran  F ps  <->  A. x  e.  A  ch ) )
 
Theoremrexrnmpt 5310* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  F  =  ( x  e.  A  |->  B )   &    |-  ( y  =  B  ->  ( ps  <->  ch ) )   =>    |-  ( A. x  e.  A  B  e.  V  ->  ( E. y  e. 
 ran  F ps  <->  E. x  e.  A  ch ) )
 
Theoremdff2 5311 Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.)
 |-  ( F : A --> B 
 <->  ( F  Fn  A  /\  F  C_  ( A  X.  B ) ) )
 
Theoremdff3im 5312* Property of a mapping. (Contributed by Jim Kingdon, 4-Jan-2019.)
 |-  ( F : A --> B  ->  ( F  C_  ( A  X.  B ) 
 /\  A. x  e.  A  E! y  x F y ) )
 
Theoremdff4im 5313* Property of a mapping. (Contributed by Jim Kingdon, 4-Jan-2019.)
 |-  ( F : A --> B  ->  ( F  C_  ( A  X.  B ) 
 /\  A. x  e.  A  E! y  e.  B  x F y ) )
 
Theoremdffo3 5314* An onto mapping expressed in terms of function values. (Contributed by NM, 29-Oct-2006.)
 |-  ( F : A -onto-> B 
 <->  ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
 
Theoremdffo4 5315* Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)
 |-  ( F : A -onto-> B 
 <->  ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  x F y ) )
 
Theoremdffo5 5316* Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)
 |-  ( F : A -onto-> B 
 <->  ( F : A --> B  /\  A. y  e.  B  E. x  x F y ) )
 
Theoremfoelrn 5317* Property of a surjective function. (Contributed by Jeff Madsen, 4-Jan-2011.)
 |-  ( ( F : A -onto-> B  /\  C  e.  B )  ->  E. x  e.  A  C  =  ( F `  x ) )
 
Theoremfoco2 5318 If a composition of two functions is surjective, then the function on the left is surjective. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  ( ( F : B
 --> C  /\  G : A
 --> B  /\  ( F  o.  G ) : A -onto-> C )  ->  F : B -onto-> C )
 
Theoremfmpt 5319* Functionality of the mapping operation. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  F  =  ( x  e.  A  |->  C )   =>    |-  ( A. x  e.  A  C  e.  B  <->  F : A --> B )
 
Theoremf1ompt 5320* Express bijection for a mapping operation. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by Mario Carneiro, 4-Dec-2016.)
 |-  F  =  ( x  e.  A  |->  C )   =>    |-  ( F : A -1-1-onto-> B  <->  ( A. x  e.  A  C  e.  B  /\  A. y  e.  B  E! x  e.  A  y  =  C )
 )
 
Theoremfmpti 5321* Functionality of the mapping operation. (Contributed by NM, 19-Mar-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
 |-  F  =  ( x  e.  A  |->  C )   &    |-  ( x  e.  A  ->  C  e.  B )   =>    |-  F : A --> B
 
Theoremfmptd 5322* Domain and codomain of the mapping operation; deduction form. (Contributed by Mario Carneiro, 13-Jan-2013.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  C )   &    |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( ph  ->  F : A --> C )
 
Theoremffnfv 5323* A function maps to a class to which all values belong. (Contributed by NM, 3-Dec-2003.)
 |-  ( F : A --> B 
 <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) )
 
Theoremffnfvf 5324 A function maps to a class to which all values belong. This version of ffnfv 5323 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 28-Sep-2006.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  F/_ x F   =>    |-  ( F : A --> B 
 <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) )
 
Theoremfnfvrnss 5325* An upper bound for range determined by function values. (Contributed by NM, 8-Oct-2004.)
 |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B )  ->  ran  F  C_  B )
 
Theoremrnmptss 5326* The range of an operation given by the "maps to" notation as a subset. (Contributed by Thierry Arnoux, 24-Sep-2017.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( A. x  e.  A  B  e.  C  ->  ran 
 F  C_  C )
 
Theoremfmpt2d 5327* Domain and codomain of the mapping operation; deduction form. (Contributed by NM, 27-Dec-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )   &    |-  ( ( ph  /\  y  e.  A )  ->  ( F `  y )  e.  C )   =>    |-  ( ph  ->  F : A --> C )
 
Theoremffvresb 5328* A necessary and sufficient condition for a restricted function. (Contributed by Mario Carneiro, 14-Nov-2013.)
 |-  ( Fun  F  ->  ( ( F  |`  A ) : A --> B  <->  A. x  e.  A  ( x  e.  dom  F 
 /\  ( F `  x )  e.  B ) ) )
 
Theoremf1oresrab 5329* Build a bijection between restricted abstract builders, given a bijection between the base classes, deduction version. (Contributed by Thierry Arnoux, 17-Aug-2018.)
 |-  F  =  ( x  e.  A  |->  C )   &    |-  ( ph  ->  F : A
 -1-1-onto-> B )   &    |-  ( ( ph  /\  x  e.  A  /\  y  =  C )  ->  ( ch  <->  ps ) )   =>    |-  ( ph  ->  ( F  |`  { x  e.  A  |  ps }
 ) : { x  e.  A  |  ps } -1-1-onto-> {
 y  e.  B  |  ch } )
 
Theoremfmptco 5330* Composition of two functions expressed as ordered-pair class abstractions. If  F has the equation ( x + 2 ) and  G the equation ( 3 * z ) then  ( G  o.  F
) has the equation ( 3 * ( x + 2 ) ) . (Contributed by FL, 21-Jun-2012.) (Revised by Mario Carneiro, 24-Jul-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  R  e.  B )   &    |-  ( ph  ->  F  =  ( x  e.  A  |->  R ) )   &    |-  ( ph  ->  G  =  ( y  e.  B  |->  S ) )   &    |-  (
 y  =  R  ->  S  =  T )   =>    |-  ( ph  ->  ( G  o.  F )  =  ( x  e.  A  |->  T ) )
 
Theoremfmptcof 5331* Version of fmptco 5330 where  ph needn't be distinct from  x. (Contributed by NM, 27-Dec-2014.)
 |-  ( ph  ->  A. x  e.  A  R  e.  B )   &    |-  ( ph  ->  F  =  ( x  e.  A  |->  R ) )   &    |-  ( ph  ->  G  =  ( y  e.  B  |->  S ) )   &    |-  ( y  =  R  ->  S  =  T )   =>    |-  ( ph  ->  ( G  o.  F )  =  ( x  e.  A  |->  T ) )
 
Theoremfmptcos 5332* Composition of two functions expressed as mapping abstractions. (Contributed by NM, 22-May-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( ph  ->  A. x  e.  A  R  e.  B )   &    |-  ( ph  ->  F  =  ( x  e.  A  |->  R ) )   &    |-  ( ph  ->  G  =  ( y  e.  B  |->  S ) )   =>    |-  ( ph  ->  ( G  o.  F )  =  ( x  e.  A  |->  [_ R  /  y ]_ S ) )
 
Theoremfcompt 5333* Express composition of two functions as a maps-to applying both in sequence. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
 |-  ( ( A : D
 --> E  /\  B : C
 --> D )  ->  ( A  o.  B )  =  ( x  e.  C  |->  ( A `  ( B `
  x ) ) ) )
 
Theoremfcoconst 5334 Composition with a constant function. (Contributed by Stefan O'Rear, 11-Mar-2015.)
 |-  ( ( F  Fn  X  /\  Y  e.  X )  ->  ( F  o.  ( I  X.  { Y } ) )  =  ( I  X.  {
 ( F `  Y ) } ) )
 
Theoremfsn 5335 A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 10-Dec-2003.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( F : { A } --> { B }  <->  F  =  { <. A ,  B >. } )
 
Theoremfsng 5336 A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 26-Oct-2012.)
 |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( F : { A } --> { B } 
 <->  F  =  { <. A ,  B >. } )
 )
 
Theoremfsn2 5337 A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by NM, 19-May-2004.)
 |-  A  e.  _V   =>    |-  ( F : { A } --> B  <->  ( ( F `
  A )  e.  B  /\  F  =  { <. A ,  ( F `  A ) >. } ) )
 
Theoremxpsng 5338 The cross product of two singletons. (Contributed by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( { A }  X.  { B }
 )  =  { <. A ,  B >. } )
 
Theoremxpsn 5339 The cross product of two singletons. (Contributed by NM, 4-Nov-2006.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( { A }  X.  { B } )  =  { <. A ,  B >. }
 
Theoremdfmpt 5340 Alternate definition for the "maps to" notation df-mpt 3820 (although it requires that  B be a set). (Contributed by NM, 24-Aug-2010.) (Revised by Mario Carneiro, 30-Dec-2016.)
 |-  B  e.  _V   =>    |-  ( x  e.  A  |->  B )  = 
 U_ x  e.  A  { <. x ,  B >. }
 
Theoremfnasrn 5341 A function expressed as the range of another function. (Contributed by Mario Carneiro, 22-Jun-2013.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
 |-  B  e.  _V   =>    |-  ( x  e.  A  |->  B )  = 
 ran  ( x  e.  A  |->  <. x ,  B >. )
 
Theoremdfmptg 5342 Alternate definition for the "maps to" notation df-mpt 3820 (which requires that  B be a set). (Contributed by Jim Kingdon, 9-Jan-2019.)
 |-  ( A. x  e.  A  B  e.  V  ->  ( x  e.  A  |->  B )  =  U_ x  e.  A  { <. x ,  B >. } )
 
Theoremfnasrng 5343 A function expressed as the range of another function. (Contributed by Jim Kingdon, 9-Jan-2019.)
 |-  ( A. x  e.  A  B  e.  V  ->  ( x  e.  A  |->  B )  =  ran  ( x  e.  A  |->  <. x ,  B >. ) )
 
Theoremressnop0 5344 If  A is not in  C, then the restriction of a singleton of  <. A ,  B >. to  C is null. (Contributed by Scott Fenton, 15-Apr-2011.)
 |-  ( -.  A  e.  C  ->  ( { <. A ,  B >. }  |`  C )  =  (/) )
 
Theoremfpr 5345 A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( A  =/=  B  ->  { <. A ,  C >. ,  <. B ,  D >. } : { A ,  B } --> { C ,  D } )
 
Theoremfprg 5346 A function with a domain of two elements. (Contributed by FL, 2-Feb-2014.)
 |-  ( ( ( A  e.  E  /\  B  e.  F )  /\  ( C  e.  G  /\  D  e.  H )  /\  A  =/=  B ) 
 ->  { <. A ,  C >. ,  <. B ,  D >. } : { A ,  B } --> { C ,  D } )
 
Theoremftpg 5347 A function with a domain of three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
 |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W )  /\  ( A  e.  F  /\  B  e.  G  /\  C  e.  H )  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z ) )  ->  { <. X ,  A >. ,  <. Y ,  B >. ,  <. Z ,  C >. } : { X ,  Y ,  Z } --> { A ,  B ,  C }
 )
 
Theoremftp 5348 A function with a domain of three elements. (Contributed by Stefan O'Rear, 17-Oct-2014.) (Proof shortened by Alexander van der Vekens, 23-Jan-2018.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  X  e.  _V   &    |-  Y  e.  _V   &    |-  Z  e.  _V   &    |-  A  =/=  B   &    |-  A  =/=  C   &    |-  B  =/=  C   =>    |-  { <. A ,  X >. ,  <. B ,  Y >. ,  <. C ,  Z >. } : { A ,  B ,  C } --> { X ,  Y ,  Z }
 
Theoremfnressn 5349 A function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)
 |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( F  |`  { B } )  =  { <. B ,  ( F `
  B ) >. } )
 
Theoremfressnfv 5350 The value of a function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)
 |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F  |`  { B } ) : { B } --> C  <->  ( F `  B )  e.  C ) )
 
Theoremfvconst 5351 The value of a constant function. (Contributed by NM, 30-May-1999.)
 |-  ( ( F : A
 --> { B }  /\  C  e.  A )  ->  ( F `  C )  =  B )
 
Theoremfmptsn 5352* Express a singleton function in maps-to notation. (Contributed by NM, 6-Jun-2006.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear, 28-Feb-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  { <. A ,  B >. }  =  ( x  e.  { A }  |->  B ) )
 
Theoremfmptap 5353* Append an additional value to a function. (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( R  u.  { A } )  =  S   &    |-  ( x  =  A  ->  C  =  B )   =>    |-  ( ( x  e.  R  |->  C )  u. 
 { <. A ,  B >. } )  =  ( x  e.  S  |->  C )
 
Theoremfmptapd 5354* Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.)
 |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  ( R  u.  { A } )  =  S )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  C  =  B )   =>    |-  ( ph  ->  ( ( x  e.  R  |->  C )  u.  { <. A ,  B >. } )  =  ( x  e.  S  |->  C ) )
 
Theoremfmptpr 5355* Express a pair function in maps-to notation. (Contributed by Thierry Arnoux, 3-Jan-2017.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  D  e.  Y )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  E  =  C )   &    |-  ( ( ph  /\  x  =  B )  ->  E  =  D )   =>    |-  ( ph  ->  { <. A ,  C >. ,  <. B ,  D >. }  =  ( x  e.  { A ,  B }  |->  E ) )
 
Theoremfvresi 5356 The value of a restricted identity function. (Contributed by NM, 19-May-2004.)
 |-  ( B  e.  A  ->  ( (  _I  |`  A ) `
  B )  =  B )
 
Theoremfvunsng 5357 Remove an ordered pair not participating in a function value. (Contributed by Jim Kingdon, 7-Jan-2019.)
 |-  ( ( D  e.  V  /\  B  =/=  D )  ->  ( ( A  u.  { <. B ,  C >. } ) `  D )  =  ( A `  D ) )
 
Theoremfvsn 5358 The value of a singleton of an ordered pair is the second member. (Contributed by NM, 12-Aug-1994.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( { <. A ,  B >. } `  A )  =  B
 
Theoremfvsng 5359 The value of a singleton of an ordered pair is the second member. (Contributed by NM, 26-Oct-2012.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( { <. A ,  B >. } `  A )  =  B )
 
Theoremfvsnun1 5360 The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 5361. (Contributed by NM, 23-Sep-2007.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  G  =  ( { <. A ,  B >. }  u.  ( F  |`  ( C  \  { A } ) ) )   =>    |-  ( G `  A )  =  B
 
Theoremfvsnun2 5361 The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 5360. (Contributed by NM, 23-Sep-2007.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  G  =  ( { <. A ,  B >. }  u.  ( F  |`  ( C  \  { A } ) ) )   =>    |-  ( D  e.  ( C  \  { A }
 )  ->  ( G `  D )  =  ( F `  D ) )
 
Theoremfsnunf 5362 Adjoining a point to a function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
 |-  ( ( F : S
 --> T  /\  ( X  e.  V  /\  -.  X  e.  S )  /\  Y  e.  T ) 
 ->  ( F  u.  { <. X ,  Y >. } ) : ( S  u.  { X }
 ) --> T )
 
Theoremfsnunfv 5363 Recover the added point from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by NM, 18-May-2017.)
 |-  ( ( X  e.  V  /\  Y  e.  W  /\  -.  X  e.  dom  F )  ->  ( ( F  u.  { <. X ,  Y >. } ) `  X )  =  Y )
 
Theoremfsnunres 5364 Recover the original function from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
 |-  ( ( F  Fn  S  /\  -.  X  e.  S )  ->  ( ( F  u.  { <. X ,  Y >. } )  |`  S )  =  F )
 
Theoremfvpr1 5365 The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)
 |-  A  e.  _V   &    |-  C  e.  _V   =>    |-  ( A  =/=  B  ->  ( { <. A ,  C >. ,  <. B ,  D >. } `  A )  =  C )
 
Theoremfvpr2 5366 The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)
 |-  B  e.  _V   &    |-  D  e.  _V   =>    |-  ( A  =/=  B  ->  ( { <. A ,  C >. ,  <. B ,  D >. } `  B )  =  D )
 
Theoremfvpr1g 5367 The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
 |-  ( ( A  e.  V  /\  C  e.  W  /\  A  =/=  B ) 
 ->  ( { <. A ,  C >. ,  <. B ,  D >. } `  A )  =  C )
 
Theoremfvpr2g 5368 The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
 |-  ( ( B  e.  V  /\  D  e.  W  /\  A  =/=  B ) 
 ->  ( { <. A ,  C >. ,  <. B ,  D >. } `  B )  =  D )
 
Theoremfvtp1g 5369 The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
 |-  ( ( ( A  e.  V  /\  D  e.  W )  /\  ( A  =/=  B  /\  A  =/=  C ) )  ->  ( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  A )  =  D )
 
Theoremfvtp2g 5370 The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
 |-  ( ( ( B  e.  V  /\  E  e.  W )  /\  ( A  =/=  B  /\  B  =/=  C ) )  ->  ( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  B )  =  E )
 
Theoremfvtp3g 5371 The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
 |-  ( ( ( C  e.  V  /\  F  e.  W )  /\  ( A  =/=  C  /\  B  =/=  C ) )  ->  ( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  C )  =  F )
 
Theoremfvtp1 5372 The first value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
 |-  A  e.  _V   &    |-  D  e.  _V   =>    |-  ( ( A  =/=  B 
 /\  A  =/=  C )  ->  ( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  A )  =  D )
 
Theoremfvtp2 5373 The second value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
 |-  B  e.  _V   &    |-  E  e.  _V   =>    |-  ( ( A  =/=  B 
 /\  B  =/=  C )  ->  ( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  B )  =  E )
 
Theoremfvtp3 5374 The third value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
 |-  C  e.  _V   &    |-  F  e.  _V   =>    |-  ( ( A  =/=  C 
 /\  B  =/=  C )  ->  ( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  C )  =  F )
 
Theoremfvconst2g 5375 The value of a constant function. (Contributed by NM, 20-Aug-2005.)
 |-  ( ( B  e.  D  /\  C  e.  A )  ->  ( ( A  X.  { B }
 ) `  C )  =  B )
 
Theoremfconst2g 5376 A constant function expressed as a cross product. (Contributed by NM, 27-Nov-2007.)
 |-  ( B  e.  C  ->  ( F : A --> { B }  <->  F  =  ( A  X.  { B }
 ) ) )
 
Theoremfvconst2 5377 The value of a constant function. (Contributed by NM, 16-Apr-2005.)
 |-  B  e.  _V   =>    |-  ( C  e.  A  ->  ( ( A  X.  { B }
 ) `  C )  =  B )
 
Theoremfconst2 5378 A constant function expressed as a cross product. (Contributed by NM, 20-Aug-1999.)
 |-  B  e.  _V   =>    |-  ( F : A
 --> { B }  <->  F  =  ( A  X.  { B }
 ) )
 
Theoremfconstfvm 5379* A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5378. (Contributed by Jim Kingdon, 8-Jan-2019.)
 |-  ( E. y  y  e.  A  ->  ( F : A --> { B } 
 <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B ) ) )
 
Theoremfconst3m 5380* Two ways to express a constant function. (Contributed by Jim Kingdon, 8-Jan-2019.)
 |-  ( E. x  x  e.  A  ->  ( F : A --> { B } 
 <->  ( F  Fn  A  /\  A  C_  ( `' F " { B }
 ) ) ) )
 
Theoremfconst4m 5381* Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)
 |-  ( E. x  x  e.  A  ->  ( F : A --> { B } 
 <->  ( F  Fn  A  /\  ( `' F " { B } )  =  A ) ) )
 
Theoremresfunexg 5382 The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. (Contributed by NM, 7-Apr-1995.) (Revised by Mario Carneiro, 22-Jun-2013.)
 |-  ( ( Fun  A  /\  B  e.  C ) 
 ->  ( A  |`  B )  e.  _V )
 
Theoremfnex 5383 If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of resfunexg 5382. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 |-  ( ( F  Fn  A  /\  A  e.  B )  ->  F  e.  _V )
 
Theoremfunex 5384 If the domain of a function exists, so the function. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of fnex 5383. (Note: Any resemblance between F.U.N.E.X. and "Have You Any Eggs" is purely a coincidence originated by Swedish chefs.) (Contributed by NM, 11-Nov-1995.)
 |-  ( ( Fun  F  /\  dom  F  e.  B )  ->  F  e.  _V )
 
Theoremopabex 5385* Existence of a function expressed as class of ordered pairs. (Contributed by NM, 21-Jul-1996.)
 |-  A  e.  _V   &    |-  ( x  e.  A  ->  E* y ph )   =>    |-  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  e.  _V
 
Theoremmptexg 5386* If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( A  e.  V  ->  ( x  e.  A  |->  B )  e.  _V )
 
Theoremmptex 5387* If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by NM, 22-Apr-2005.) (Revised by Mario Carneiro, 20-Dec-2013.)
 |-  A  e.  _V   =>    |-  ( x  e.  A  |->  B )  e. 
 _V
 
Theoremfex 5388 If the domain of a mapping is a set, the function is a set. (Contributed by NM, 3-Oct-1999.)
 |-  ( ( F : A
 --> B  /\  A  e.  C )  ->  F  e.  _V )
 
Theoremeufnfv 5389* A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 E! f ( f  Fn  A  /\  A. x  e.  A  (
 f `  x )  =  B )
 
Theoremfunfvima 5390 A function's value in a preimage belongs to the image. (Contributed by NM, 23-Sep-2003.)
 |-  ( ( Fun  F  /\  B  e.  dom  F )  ->  ( B  e.  A  ->  ( F `  B )  e.  ( F " A ) ) )
 
Theoremfunfvima2 5391 A function's value in an included preimage belongs to the image. (Contributed by NM, 3-Feb-1997.)
 |-  ( ( Fun  F  /\  A  C_  dom  F ) 
 ->  ( B  e.  A  ->  ( F `  B )  e.  ( F " A ) ) )
 
Theoremfunfvima3 5392 A class including a function contains the function's value in the image of the singleton of the argument. (Contributed by NM, 23-Mar-2004.)
 |-  ( ( Fun  F  /\  F  C_  G )  ->  ( A  e.  dom  F 
 ->  ( F `  A )  e.  ( G " { A } )
 ) )
 
Theoremfnfvima 5393 The function value of an operand in a set is contained in the image of that set, using the  Fn abbreviation. (Contributed by Stefan O'Rear, 10-Mar-2015.)
 |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S ) 
 ->  ( F `  X )  e.  ( F " S ) )
 
Theoremrexima 5394* Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)
 |-  ( x  =  ( F `  y ) 
 ->  ( ph  <->  ps ) )   =>    |-  ( ( F  Fn  A  /\  B  C_  A )  ->  ( E. x  e.  ( F " B ) ph  <->  E. y  e.  B  ps ) )
 
Theoremralima 5395* Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)
 |-  ( x  =  ( F `  y ) 
 ->  ( ph  <->  ps ) )   =>    |-  ( ( F  Fn  A  /\  B  C_  A )  ->  ( A. x  e.  ( F " B ) ph  <->  A. y  e.  B  ps ) )
 
Theoremidref 5396* TODO: This is the same as issref 4707 (which has a much longer proof). Should we replace issref 4707 with this one? - NM 9-May-2016.

Two ways to state a relation is reflexive. (Adapted from Tarski.) (Contributed by FL, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) (Proof modification is discouraged.)

 |-  ( (  _I  |`  A ) 
 C_  R  <->  A. x  e.  A  x R x )
 
Theoremelabrex 5397* Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.)
 |-  B  e.  _V   =>    |-  ( x  e.  A  ->  B  e.  { y  |  E. x  e.  A  y  =  B } )
 
Theoremabrexco 5398* Composition of two image maps  C ( y ) and 
B ( w ). (Contributed by NM, 27-May-2013.)
 |-  B  e.  _V   &    |-  (
 y  =  B  ->  C  =  D )   =>    |-  { x  |  E. y  e.  { z  |  E. w  e.  A  z  =  B } x  =  C }  =  { x  |  E. w  e.  A  x  =  D }
 
Theoremimaiun 5399* The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( A " U_ x  e.  B  C )  = 
 U_ x  e.  B  ( A " C )
 
Theoremimauni 5400* The image of a union is the indexed union of the images. Theorem 3K(a) of [Enderton] p. 50. (Contributed by NM, 9-Aug-2004.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)
 |-  ( A " U. B )  =  U_ x  e.  B  ( A " x )
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