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Theorem List for Metamath Proof Explorer - 6001-6100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoreminpreima 6001 Preimage of an intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jun-2016.)
 |-  ( Fun  F  ->  ( `' F " ( A  i^i  B ) )  =  ( ( `' F " A )  i^i  ( `' F " B ) ) )
 
Theoremdifpreima 6002 Preimage of a difference. (Contributed by Mario Carneiro, 14-Jun-2016.)
 |-  ( Fun  F  ->  ( `' F " ( A 
 \  B ) )  =  ( ( `' F " A ) 
 \  ( `' F " B ) ) )
 
Theoremrespreima 6003 The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( Fun  F  ->  ( `' ( F  |`  B )
 " A )  =  ( ( `' F " A )  i^i  B ) )
 
Theoremiinpreima 6004* Preimage of an intersection. (Contributed by FL, 16-Apr-2012.)
 |-  ( ( Fun  F  /\  A  =/=  (/) )  ->  ( `' F " |^|_ x  e.  A  B )  = 
 |^|_ x  e.  A  ( `' F " B ) )
 
Theoremintpreima 6005* Preimage of an intersection. (Contributed by FL, 28-Apr-2012.)
 |-  ( ( Fun  F  /\  A  =/=  (/) )  ->  ( `' F " |^| A )  =  |^|_ x  e.  A  ( `' F " x ) )
 
Theoremfimacnv 6006 The preimage of the codomain of a mapping is the mapping's domain. (Contributed by FL, 25-Jan-2007.)
 |-  ( F : A --> B  ->  ( `' F " B )  =  A )
 
TheoremsuppssOLD 6007* Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) Obsolete version of suppss 6922 as of 28-May-2019. (New usage is discouraged.)
 |-  ( ph  ->  F : A --> B )   &    |-  (
 ( ph  /\  k  e.  ( A  \  W ) )  ->  ( F `
  k )  =  Z )   =>    |-  ( ph  ->  ( `' F " ( _V  \  { Z } )
 )  C_  W )
 
TheoremsuppssrOLD 6008 A function is zero outside its support. (Contributed by Mario Carneiro, 19-Dec-2014.) Obsolete version of suppssr 6923 as of 28-May-2019. (New usage is discouraged.)
 |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  ( `' F " ( _V  \  { Z } ) )  C_  W )   =>    |-  ( ( ph  /\  X  e.  ( A  \  W ) )  ->  ( F `
  X )  =  Z )
 
Theoremfnopfv 6009 Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 30-Sep-2004.)
 |-  ( ( F  Fn  A  /\  B  e.  A )  ->  <. B ,  ( F `  B ) >.  e.  F )
 
Theoremfvelrn 6010 A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.)
 |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  ( F `  A )  e.  ran  F )
 
Theoremfnfvelrn 6011 A function's value belongs to its range. (Contributed by NM, 15-Oct-1996.)
 |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( F `  B )  e.  ran  F )
 
Theoremffvelrn 6012 A function's value belongs to its codomain. (Contributed by NM, 12-Aug-1999.)
 |-  ( ( F : A
 --> B  /\  C  e.  A )  ->  ( F `
  C )  e.  B )
 
Theoremffvelrni 6013 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 6014 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 6015 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 6016* 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 6017* 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 6018* 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 6019* 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 6020* 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 ) ) )
 
Theoremeldmrexrnb 6021* For any element in the domain of a function, there is an element in the range of the function which is the value of the function at that element. Because of the definition df-fv 5589 of the value of a function, the theorem is only valid in general if the empty set is not contained in the range of the function (the implication "to the right" is always valid). Indeed, with the definition df-fv 5589 of the value of a function,  ( F `  Y )  =  (/) may mean that the value of  F at  Y is the empty set or that  F is not defined at  Y. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
 |-  ( ( Fun  F  /\  (/)  e/  ran  F ) 
 ->  ( Y  e.  dom  F  <->  E. x  e.  ran  F  x  =  ( F `
  Y ) ) )
 
Theoremfvcofneq 6022* The values of two function compositions are equal if the values of the composed functions are pairwise equal. (Contributed by AV, 26-Jan-2019.)
 |-  ( ( G  Fn  A  /\  K  Fn  B )  ->  ( ( X  e.  ( A  i^i  B )  /\  ( G `
  X )  =  ( K `  X )  /\  A. x  e.  ( ran  G  i^i  ran 
 K ) ( F `
  x )  =  ( H `  x ) )  ->  ( ( F  o.  G ) `
  X )  =  ( ( H  o.  K ) `  X ) ) )
 
Theoremralrnmpt 6023* 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 6024* 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 ) )
 
Theoremf0cli 6025 Unconditional closure of a function when the range includes the empty set. (Contributed by Mario Carneiro, 12-Sep-2013.)
 |-  F : A --> B   &    |-  (/)  e.  B   =>    |-  ( F `  C )  e.  B
 
Theoremdff2 6026 Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.)
 |-  ( F : A --> B 
 <->  ( F  Fn  A  /\  F  C_  ( A  X.  B ) ) )
 
Theoremdff3 6027* Alternate definition of a mapping. (Contributed by NM, 20-Mar-2007.)
 |-  ( F : A --> B 
 <->  ( F  C_  ( A  X.  B )  /\  A. x  e.  A  E! y  x F y ) )
 
Theoremdff4 6028* Alternate definition of a mapping. (Contributed by NM, 20-Mar-2007.)
 |-  ( F : A --> B 
 <->  ( F  C_  ( A  X.  B )  /\  A. x  e.  A  E! y  e.  B  x F y ) )
 
Theoremdffo3 6029* 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 6030* 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 6031* 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 ) )
 
Theoremexfo 6032* A relation equivalent to the existence of an onto mapping. The right-hand  f is not necessarily a function. (Contributed by NM, 20-Mar-2007.)
 |-  ( E. f  f : A -onto-> B  <->  E. f ( A. x  e.  A  E! y  e.  B  x f y  /\  A. x  e.  B  E. y  e.  A  y f x ) )
 
Theoremfoelrn 6033* 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 6034 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 6035* 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 6036* 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 6037* 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 6038* 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 )
 
Theoremfmptdf 6039* A version of fmptd 6038 using bound-variable hypothesis instead of a distinct variable condition for  ph. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |- 
 F/ x ph   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  C )   &    |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( ph  ->  F : A --> C )
 
Theoremffnfv 6040* 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 6041 A function maps to a class to which all values belong. This version of ffnfv 6040 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 6042* 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 6043* 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 6044* 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 6045* 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 6046* 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 6047* 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 6048* Version of fmptco 6047 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 6049* 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 6050* 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 6051 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 6052 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 6053 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 6054 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 6055 The Cartesian product of two singletons. (Contributed by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( { A }  X.  { B }
 )  =  { <. A ,  B >. } )
 
Theoremxpsn 6056 The Cartesian product of two singletons. (Contributed by NM, 4-Nov-2006.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( { A }  X.  { B } )  =  { <. A ,  B >. }
 
Theoremf1o2sn 6057 A singleton with a nested ordered pair is a 1-1 function of the cartesian product of two singleton onto a singleton. (Contributed by AV, 15-Aug-2019.)
 |-  ( ( E  e.  V  /\  X  e.  W )  ->  { <. <. E ,  E >. ,  X >. } : ( { E }  X.  { E }
 )
 -1-1-onto-> { X } )
 
Theoremresidpr 6058 Restriction of the identity to a pair. (Contributed by AV, 11-Dec-2018.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  (  _I  |`  { A ,  B } )  =  { <. A ,  A >. ,  <. B ,  B >. } )
 
Theoremdfmpt 6059 Alternate definition for the "maps to" notation df-mpt 4502 (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 6060 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 >. )
 
Theoremressnop0 6061 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 6062 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 6063 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 6064 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 6065 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 6066 A function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)
 |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( F  |`  { B } )  =  { <. B ,  ( F `
  B ) >. } )
 
Theoremfunressn 6067 A function restricted to a singleton. (Contributed by Mario Carneiro, 16-Nov-2014.)
 |-  ( Fun  F  ->  ( F  |`  { B } )  C_  { <. B ,  ( F `  B ) >. } )
 
Theoremfressnfv 6068 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 ) )
 
Theoremfvrnressn 6069 If the value of a function is in the range of the function restricted to the singleton containing the argument, then the value of the function is in the range of the function. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
 |-  ( X  e.  V  ->  ( ( F `  X )  e.  ran  ( F  |`  { X } )  ->  ( F `
  X )  e. 
 ran  F ) )
 
Theoremfvressn 6070 The value of a function restricted to the singleton containing the argument equals the value of the function for this argument. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
 |-  ( X  e.  V  ->  ( ( F  |`  { X } ) `  X )  =  ( F `  X ) )
 
Theoremfvn0fvelrn 6071 If the value of a function is not null, the value is an element of the range of the function. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
 |-  ( ( F `  X )  =/=  (/)  ->  ( F `  X )  e. 
 ran  F )
 
Theoremfvconst 6072 The value of a constant function. (Contributed by NM, 30-May-1999.)
 |-  ( ( F : A
 --> { B }  /\  C  e.  A )  ->  ( F `  C )  =  B )
 
Theoremfnsnb 6073 A function whose domain is a singleton can be represented as a singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) Revised to add reverse implication. (Revised by NM, 29-Dec-2018.)
 |-  A  e.  _V   =>    |-  ( F  Fn  { A }  <->  F  =  { <. A ,  ( F `
  A ) >. } )
 
Theoremfmptsn 6074* 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 ) )
 
Theoremfmptsng 6075* Express a singleton function in maps-to notation. Version of fmptsn 6074 allowing the mapping value to depend on the mapping variable (usual case). (Contributed by AV, 27-Feb-2019.)
 |-  ( x  =  A  ->  B  =  C )   =>    |-  ( ( A  e.  V  /\  C  e.  W )  ->  { <. A ,  C >. }  =  ( x  e.  { A }  |->  B ) )
 
Theoremfmptsnd 6076* Express a singleton function in maps-to notation. Deduction form of fmptsng 6075. (Contributed by AV, 4-Aug-2019.)
 |-  ( ( ph  /\  x  =  A )  ->  B  =  C )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  C  e.  W )   =>    |-  ( ph  ->  { <. A ,  C >. }  =  ( x  e.  { A }  |->  B ) )
 
Theoremfmptap 6077* 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 6078* 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 6079* 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 6080 The value of a restricted identity function. (Contributed by NM, 19-May-2004.)
 |-  ( B  e.  A  ->  ( (  _I  |`  A ) `
  B )  =  B )
 
Theoremfninfp 6081* Express the class of fixed points of a function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( F  Fn  A  ->  dom  ( F  i^i  _I  )  =  { x  e.  A  |  ( F `
  x )  =  x } )
 
Theoremfnelfp 6082 Property of a fixed point of a function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( X  e.  dom  ( F  i^i  _I  ) 
 <->  ( F `  X )  =  X )
 )
 
Theoremfndifnfp 6083* Express the class of non-fixed points of a function. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( F  Fn  A  ->  dom  ( F  \  _I  )  =  { x  e.  A  |  ( F `  x )  =/=  x } )
 
Theoremfnelnfp 6084 Property of a non-fixed point of a function. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( X  e.  dom  ( F  \  _I  ) 
 <->  ( F `  X )  =/=  X ) )
 
Theoremfnnfpeq0 6085 A function is the identity iff it moves no points. (Contributed by Stefan O'Rear, 25-Aug-2015.)
 |-  ( F  Fn  A  ->  ( dom  ( F 
 \  _I  )  =  (/) 
 <->  F  =  (  _I  |`  A ) ) )
 
Theoremfvunsn 6086 Remove an ordered pair not participating in a function value. (Contributed by NM, 1-Oct-2013.) (Revised by Mario Carneiro, 28-May-2014.)
 |-  ( B  =/=  D  ->  ( ( A  u.  {
 <. B ,  C >. } ) `  D )  =  ( A `  D ) )
 
Theoremfvsn 6087 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 6088 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 6089 The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 6090. (Contributed by NM, 23-Sep-2007.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  G  =  ( { <. A ,  B >. }  u.  ( F  |`  ( C  \  { A } ) ) )   =>    |-  ( G `  A )  =  B
 
Theoremfvsnun2 6090 The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 6089. (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 ) )
 
Theoremfnsnsplit 6091 Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.)
 |-  ( ( F  Fn  A  /\  X  e.  A )  ->  F  =  ( ( F  |`  ( A 
 \  { X }
 ) )  u.  { <. X ,  ( F `
  X ) >. } ) )
 
Theoremfsnunf 6092 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 )
 
Theoremfsnunf2 6093 Adjoining a point to a punctured function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
 |-  ( ( F :
 ( S  \  { X } ) --> T  /\  X  e.  S  /\  Y  e.  T )  ->  ( F  u.  { <. X ,  Y >. } ) : S --> T )
 
Theoremfsnunfv 6094 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 6095 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 )
 
Theoremfunresdfunsn 6096 Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in the function itself. (Contributed by AV, 2-Dec-2018.)
 |-  ( ( Fun  F  /\  X  e.  dom  F )  ->  ( ( F  |`  ( _V  \  { X } ) )  u. 
 { <. X ,  ( F `  X ) >. } )  =  F )
 
Theoremfvpr1 6097 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 6098 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 6099 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 6100 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 )
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