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Theorem List for Intuitionistic Logic Explorer - 5201-5300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfv2 5201* 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 ) }
 
Theoremdffv3g 5202* A definition of function value in terms of iota. (Contributed by Jim Kingdon, 29-Dec-2018.)
 |-  ( A  e.  V  ->  ( F `  A )  =  ( iota x x  e.  ( F
 " { A }
 ) ) )
 
Theoremdffv4g 5203* 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 4722), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  e.  V  ->  ( F `  A )  =  U. { x  |  ( F " { A } )  =  { x } } )
 
Theoremelfv 5204* 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 ) ) )
 
Theoremfveq1 5205 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
 |-  ( F  =  G  ->  ( F `  A )  =  ( G `  A ) )
 
Theoremfveq2 5206 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
 |-  ( A  =  B  ->  ( F `  A )  =  ( F `  B ) )
 
Theoremfveq1i 5207 Equality inference for function value. (Contributed by NM, 2-Sep-2003.)
 |-  F  =  G   =>    |-  ( F `  A )  =  ( G `  A )
 
Theoremfveq1d 5208 Equality deduction for function value. (Contributed by NM, 2-Sep-2003.)
 |-  ( ph  ->  F  =  G )   =>    |-  ( ph  ->  ( F `  A )  =  ( G `  A ) )
 
Theoremfveq2i 5209 Equality inference for function value. (Contributed by NM, 28-Jul-1999.)
 |-  A  =  B   =>    |-  ( F `  A )  =  ( F `  B )
 
Theoremfveq2d 5210 Equality deduction for function value. (Contributed by NM, 29-May-1999.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F `  A )  =  ( F `  B ) )
 
Theoremfveq12i 5211 Equality deduction for function value. (Contributed by FL, 27-Jun-2014.)
 |-  F  =  G   &    |-  A  =  B   =>    |-  ( F `  A )  =  ( G `  B )
 
Theoremfveq12d 5212 Equality deduction for function value. (Contributed by FL, 22-Dec-2008.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F `  A )  =  ( G `  B ) )
 
Theoremnffv 5213 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 )
 
Theoremnffvmpt1 5214* Bound-variable hypothesis builder for mapping, special case. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  F/_ x ( ( x  e.  A  |->  B ) `
  C )
 
Theoremnffvd 5215 Deduction version of bound-variable hypothesis builder nffv 5213. (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 ) )
 
Theoremfunfveu 5216* A function has one value given an argument in its domain. (Contributed by Jim Kingdon, 29-Dec-2018.)
 |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  E! y  A F y )
 
Theoremfvss 5217* The value of a function is a subset of  B if every element that could be a candidate for the value is a subset of  B. (Contributed by Mario Carneiro, 24-May-2019.)
 |-  ( A. x ( A F x  ->  x  C_  B )  ->  ( F `  A ) 
 C_  B )
 
Theoremfvssunirng 5218 The result of a function value is always a subset of the union of the range, if the input is a set. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 24-May-2019.)
 |-  ( A  e.  _V  ->  ( F `  A )  C_  U. ran  F )
 
Theoremrelfvssunirn 5219 The result of a function value is always a subset of the union of the range, even if it is invalid and thus empty. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 24-May-2019.)
 |-  ( Rel  F  ->  ( F `  A ) 
 C_  U. ran  F )
 
Theoremfunfvex 5220 The value of a function exists. A special case of Corollary 6.13 of [TakeutiZaring] p. 27. (Contributed by Jim Kingdon, 29-Dec-2018.)
 |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  ( F `  A )  e.  _V )
 
Theoremrelrnfvex 5221 If a function has a set range, then the function value exists unconditional on the domain. (Contributed by Mario Carneiro, 24-May-2019.)
 |-  ( ( Rel  F  /\  ran  F  e.  _V )  ->  ( F `  A )  e.  _V )
 
Theoremfvexg 5222 Evaluating a set function at a set exists. (Contributed by Mario Carneiro and Jim Kingdon, 28-May-2019.)
 |-  ( ( F  e.  V  /\  A  e.  W )  ->  ( F `  A )  e.  _V )
 
Theoremfvex 5223 Evaluating a set function at a set exists. (Contributed by Mario Carneiro and Jim Kingdon, 28-May-2019.)
 |-  F  e.  V   &    |-  A  e.  W   =>    |-  ( F `  A )  e.  _V
 
Theoremsefvex 5224 If a function is set-like, then the function value exists if the input does. (Contributed by Mario Carneiro, 24-May-2019.)
 |-  ( ( `' F Se  _V 
 /\  A  e.  _V )  ->  ( F `  A )  e.  _V )
 
Theoremfv3 5225* Alternate definition of the value of a function. Definition 6.11 of [TakeutiZaring] p. 26. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( F `  A )  =  { x  |  ( E. y ( x  e.  y  /\  A F y )  /\  E! y  A F y ) }
 
Theoremfvres 5226 The value of a restricted function. (Contributed by NM, 2-Aug-1994.)
 |-  ( A  e.  B  ->  ( ( F  |`  B ) `
  A )  =  ( F `  A ) )
 
Theoremfunssfv 5227 The value of a member of the domain of a subclass of a function. (Contributed by NM, 15-Aug-1994.)
 |-  ( ( Fun  F  /\  G  C_  F  /\  A  e.  dom  G ) 
 ->  ( F `  A )  =  ( G `  A ) )
 
Theoremtz6.12-1 5228* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
 |-  ( ( A F y  /\  E! y  A F y )  ->  ( F `  A )  =  y )
 
Theoremtz6.12 5229* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 10-Jul-1994.)
 |-  ( ( <. A ,  y >.  e.  F  /\  E! y <. A ,  y >.  e.  F )  ->  ( F `  A )  =  y )
 
Theoremtz6.12f 5230* Function value, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 30-Aug-1999.)
 |-  F/_ y F   =>    |-  ( ( <. A ,  y >.  e.  F  /\  E! y <. A ,  y >.  e.  F )  ->  ( F `  A )  =  y )
 
Theoremtz6.12c 5231* Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
 |-  ( E! y  A F y  ->  (
 ( F `  A )  =  y  <->  A F y ) )
 
Theoremndmfvg 5232 The value of a class outside its domain is the empty set. (Contributed by Jim Kingdon, 15-Jan-2019.)
 |-  ( ( A  e.  _V 
 /\  -.  A  e.  dom 
 F )  ->  ( F `  A )  =  (/) )
 
Theoremrelelfvdm 5233 If a function value has a member, the argument belongs to the domain. (Contributed by Jim Kingdon, 22-Jan-2019.)
 |-  ( ( Rel  F  /\  A  e.  ( F `
  B ) ) 
 ->  B  e.  dom  F )
 
Theoremnfvres 5234 The value of a non-member of a restriction is the empty set. (Contributed by NM, 13-Nov-1995.)
 |-  ( -.  A  e.  B  ->  ( ( F  |`  B ) `  A )  =  (/) )
 
Theoremnfunsn 5235 If the restriction of a class to a singleton is not a function, its value is the empty set. (Contributed by NM, 8-Aug-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( -.  Fun  ( F  |`  { A }
 )  ->  ( F `  A )  =  (/) )
 
Theorem0fv 5236 Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.)
 |-  ( (/) `  A )  =  (/)
 
Theoremcsbfv12g 5237 Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.)
 |-  ( A  e.  C  -> 
 [_ A  /  x ]_ ( F `  B )  =  ( [_ A  /  x ]_ F ` 
 [_ A  /  x ]_ B ) )
 
Theoremcsbfv2g 5238* Move class substitution in and out of a function value. (Contributed by NM, 10-Nov-2005.)
 |-  ( A  e.  C  -> 
 [_ A  /  x ]_ ( F `  B )  =  ( F ` 
 [_ A  /  x ]_ B ) )
 
Theoremcsbfvg 5239* Substitution for a function value. (Contributed by NM, 1-Jan-2006.)
 |-  ( A  e.  C  -> 
 [_ A  /  x ]_ ( F `  x )  =  ( F `  A ) )
 
Theoremfunbrfv 5240 The second argument of a binary relation on a function is the function's value. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( Fun  F  ->  ( A F B  ->  ( F `  A )  =  B ) )
 
Theoremfunopfv 5241 The second element in an ordered pair member of a function is the function's value. (Contributed by NM, 19-Jul-1996.)
 |-  ( Fun  F  ->  (
 <. A ,  B >.  e.  F  ->  ( F `  A )  =  B ) )
 
Theoremfnbrfvb 5242 Equivalence of function value and binary relation. (Contributed by NM, 19-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `
  B )  =  C  <->  B F C ) )
 
Theoremfnopfvb 5243 Equivalence of function value and ordered pair membership. (Contributed by NM, 7-Nov-1995.)
 |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `
  B )  =  C  <->  <. B ,  C >.  e.  F ) )
 
Theoremfunbrfvb 5244 Equivalence of function value and binary relation. (Contributed by NM, 26-Mar-2006.)
 |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  ( ( F `
  A )  =  B  <->  A F B ) )
 
Theoremfunopfvb 5245 Equivalence of function value and ordered pair membership. Theorem 4.3(ii) of [Monk1] p. 42. (Contributed by NM, 26-Jan-1997.)
 |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  ( ( F `
  A )  =  B  <->  <. A ,  B >.  e.  F ) )
 
Theoremfunbrfv2b 5246 Function value in terms of a binary relation. (Contributed by Mario Carneiro, 19-Mar-2014.)
 |-  ( Fun  F  ->  ( A F B  <->  ( A  e.  dom 
 F  /\  ( F `  A )  =  B ) ) )
 
Theoremdffn5im 5247* Representation of a function in terms of its values. The converse holds given the law of the excluded middle; as it is we have most of the converse via funmpt 4966 and dmmptss 4845. (Contributed by Jim Kingdon, 31-Dec-2018.)
 |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F `
  x ) ) )
 
Theoremfnrnfv 5248* The range of a function expressed as a collection of the function's values. (Contributed by NM, 20-Oct-2005.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
 |-  ( F  Fn  A  ->  ran  F  =  {
 y  |  E. x  e.  A  y  =  ( F `  x ) } )
 
Theoremfvelrnb 5249* A member of a function's range is a value of the function. (Contributed by NM, 31-Oct-1995.)
 |-  ( F  Fn  A  ->  ( B  e.  ran  F  <->  E. x  e.  A  ( F `  x )  =  B ) )
 
Theoremdfimafn 5250* Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.)
 |-  ( ( Fun  F  /\  A  C_  dom  F ) 
 ->  ( F " A )  =  { y  |  E. x  e.  A  ( F `  x )  =  y } )
 
Theoremdfimafn2 5251* Alternate definition of the image of a function as an indexed union of singletons of function values. (Contributed by Raph Levien, 20-Nov-2006.)
 |-  ( ( Fun  F  /\  A  C_  dom  F ) 
 ->  ( F " A )  =  U_ x  e.  A  { ( F `
  x ) }
 )
 
Theoremfunimass4 5252* Membership relation for the values of a function whose image is a subclass. (Contributed by Raph Levien, 20-Nov-2006.)
 |-  ( ( Fun  F  /\  A  C_  dom  F ) 
 ->  ( ( F " A )  C_  B  <->  A. x  e.  A  ( F `  x )  e.  B ) )
 
Theoremfvelima 5253* Function value in an image. Part of Theorem 4.4(iii) of [Monk1] p. 42. (Contributed by NM, 29-Apr-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( ( Fun  F  /\  A  e.  ( F
 " B ) ) 
 ->  E. x  e.  B  ( F `  x )  =  A )
 
Theoremfeqmptd 5254* Deduction form of dffn5im 5247. (Contributed by Mario Carneiro, 8-Jan-2015.)
 |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
 
Theoremfeqresmpt 5255* Express a restricted function as a mapping. (Contributed by Mario Carneiro, 18-May-2016.)
 |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  C  C_  A )   =>    |-  ( ph  ->  ( F  |`  C )  =  ( x  e.  C  |->  ( F `  x ) ) )
 
Theoremdffn5imf 5256* Representation of a function in terms of its values. (Contributed by Jim Kingdon, 31-Dec-2018.)
 |-  F/_ x F   =>    |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F `
  x ) ) )
 
Theoremfvelimab 5257* Function value in an image. (Contributed by NM, 20-Jan-2007.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by David Abernethy, 17-Dec-2011.)
 |-  ( ( F  Fn  A  /\  B  C_  A )  ->  ( C  e.  ( F " B )  <->  E. x  e.  B  ( F `  x )  =  C ) )
 
Theoremfvi 5258 The value of the identity function. (Contributed by NM, 1-May-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( A  e.  V  ->  (  _I  `  A )  =  A )
 
Theoremfniinfv 5259* The indexed intersection of a function's values is the intersection of its range. (Contributed by NM, 20-Oct-2005.)
 |-  ( F  Fn  A  -> 
 |^|_ x  e.  A  ( F `  x )  =  |^| ran  F )
 
Theoremfnsnfv 5260 Singleton of function value. (Contributed by NM, 22-May-1998.)
 |-  ( ( F  Fn  A  /\  B  e.  A )  ->  { ( F `
  B ) }  =  ( F " { B } ) )
 
Theoremfnimapr 5261 The image of a pair under a function. (Contributed by Jeff Madsen, 6-Jan-2011.)
 |-  ( ( F  Fn  A  /\  B  e.  A  /\  C  e.  A ) 
 ->  ( F " { B ,  C }
 )  =  { ( F `  B ) ,  ( F `  C ) } )
 
Theoremssimaex 5262* The existence of a subimage. (Contributed by NM, 8-Apr-2007.)
 |-  A  e.  _V   =>    |-  ( ( Fun 
 F  /\  B  C_  ( F " A ) ) 
 ->  E. x ( x 
 C_  A  /\  B  =  ( F " x ) ) )
 
Theoremssimaexg 5263* The existence of a subimage. (Contributed by FL, 15-Apr-2007.)
 |-  ( ( A  e.  C  /\  Fun  F  /\  B  C_  ( F " A ) )  ->  E. x ( x  C_  A  /\  B  =  ( F " x ) ) )
 
Theoremfunfvdm 5264 A simplified expression for the value of a function when we know it's a function. (Contributed by Jim Kingdon, 1-Jan-2019.)
 |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  ( F `  A )  =  U. ( F " { A } ) )
 
Theoremfunfvdm2 5265* The value of a function. Definition of function value in [Enderton] p. 43. (Contributed by Jim Kingdon, 1-Jan-2019.)
 |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  ( F `  A )  =  U. { y  |  A F y } )
 
Theoremfunfvdm2f 5266 The value of a function. Version of funfvdm2 5265 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 1-Jan-2019.)
 |-  F/_ y A   &    |-  F/_ y F   =>    |-  ( ( Fun 
 F  /\  A  e.  dom 
 F )  ->  ( F `  A )  = 
 U. { y  |  A F y }
 )
 
Theoremfvun1 5267 The value of a union when the argument is in the first domain. (Contributed by Scott Fenton, 29-Jun-2013.)
 |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  ( ( F  u.  G ) `  X )  =  ( F `  X ) )
 
Theoremfvun2 5268 The value of a union when the argument is in the second domain. (Contributed by Scott Fenton, 29-Jun-2013.)
 |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  B ) )  ->  ( ( F  u.  G ) `  X )  =  ( G `  X ) )
 
Theoremdmfco 5269 Domains of a function composition. (Contributed by NM, 27-Jan-1997.)
 |-  ( ( Fun  G  /\  A  e.  dom  G )  ->  ( A  e.  dom  ( F  o.  G ) 
 <->  ( G `  A )  e.  dom  F ) )
 
Theoremfvco2 5270 Value of a function composition. Similar to second part of Theorem 3H of [Enderton] p. 47. (Contributed by NM, 9-Oct-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear, 16-Oct-2014.)
 |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( ( F  o.  G ) `  X )  =  ( F `  ( G `  X ) ) )
 
Theoremfvco 5271 Value of a function composition. Similar to Exercise 5 of [TakeutiZaring] p. 28. (Contributed by NM, 22-Apr-2006.) (Proof shortened by Mario Carneiro, 26-Dec-2014.)
 |-  ( ( Fun  G  /\  A  e.  dom  G )  ->  ( ( F  o.  G ) `  A )  =  ( F `  ( G `  A ) ) )
 
Theoremfvco3 5272 Value of a function composition. (Contributed by NM, 3-Jan-2004.) (Revised by Mario Carneiro, 26-Dec-2014.)
 |-  ( ( G : A
 --> B  /\  C  e.  A )  ->  ( ( F  o.  G ) `
  C )  =  ( F `  ( G `  C ) ) )
 
Theoremfvopab3g 5273* Value of a function given by ordered-pair class abstraction. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   &    |-  ( x  e.  C  ->  E! y ph )   &    |-  F  =  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) }   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( F `
  A )  =  B  <->  ch ) )
 
Theoremfvopab3ig 5274* Value of a function given by ordered-pair class abstraction. (Contributed by NM, 23-Oct-1999.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   &    |-  ( x  e.  C  ->  E* y ph )   &    |-  F  =  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) }   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ch  ->  ( F `  A )  =  B ) )
 
Theoremfvmptss2 5275* A mapping always evaluates to a subset of the substituted expression in the mapping, even if this is a proper class, or we are out of the domain. (Contributed by Mario Carneiro, 13-Feb-2015.) (Revised by Mario Carneiro, 3-Jul-2019.)
 |-  ( x  =  D  ->  B  =  C )   &    |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( F `  D )  C_  C
 
Theoremfvmptg 5276* Value of a function given in maps-to notation. (Contributed by NM, 2-Oct-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( x  =  A  ->  B  =  C )   &    |-  F  =  ( x  e.  D  |->  B )   =>    |-  ( ( A  e.  D  /\  C  e.  R )  ->  ( F `  A )  =  C )
 
Theoremfvmpt 5277* Value of a function given in maps-to notation. (Contributed by NM, 17-Aug-2011.)
 |-  ( x  =  A  ->  B  =  C )   &    |-  F  =  ( x  e.  D  |->  B )   &    |-  C  e.  _V   =>    |-  ( A  e.  D  ->  ( F `  A )  =  C )
 
Theoremfvmpts 5278* Value of a function given in maps-to notation, using explicit class substitution. (Contributed by Scott Fenton, 17-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  F  =  ( x  e.  C  |->  B )   =>    |-  ( ( A  e.  C  /\  [_ A  /  x ]_ B  e.  V ) 
 ->  ( F `  A )  =  [_ A  /  x ]_ B )
 
Theoremfvmpt3 5279* Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( x  =  A  ->  B  =  C )   &    |-  F  =  ( x  e.  D  |->  B )   &    |-  ( x  e.  D  ->  B  e.  V )   =>    |-  ( A  e.  D  ->  ( F `  A )  =  C )
 
Theoremfvmpt3i 5280* Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Mario Carneiro, 11-Sep-2015.)
 |-  ( x  =  A  ->  B  =  C )   &    |-  F  =  ( x  e.  D  |->  B )   &    |-  B  e.  _V   =>    |-  ( A  e.  D  ->  ( F `  A )  =  C )
 
Theoremfvmptd 5281* Deduction version of fvmpt 5277. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( ph  ->  F  =  ( x  e.  D  |->  B ) )   &    |-  (
 ( ph  /\  x  =  A )  ->  B  =  C )   &    |-  ( ph  ->  A  e.  D )   &    |-  ( ph  ->  C  e.  V )   =>    |-  ( ph  ->  ( F `  A )  =  C )
 
Theoremfvmpt2 5282* Value of a function given by the "maps to" notation. (Contributed by FL, 21-Jun-2010.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( ( x  e.  A  /\  B  e.  C )  ->  ( F `
  x )  =  B )
 
Theoremfvmptssdm 5283* If all the values of the mapping are subsets of a class  C, then so is any evaluation of the mapping at a value in the domain of the mapping. (Contributed by Jim Kingdon, 3-Jan-2018.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( ( D  e.  dom 
 F  /\  A. x  e.  A  B  C_  C )  ->  ( F `  D )  C_  C )
 
Theoremmptfvex 5284* Sufficient condition for a maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( ( A. x  B  e.  V  /\  C  e.  W )  ->  ( F `  C )  e.  _V )
 
Theoremfvmpt2d 5285* Deduction version of fvmpt2 5282. (Contributed by Thierry Arnoux, 8-Dec-2016.)
 |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   =>    |-  ( ( ph  /\  x  e.  A )  ->  ( F `  x )  =  B )
 
Theoremfvmptdf 5286* Alternate deduction version of fvmpt 5277, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |-  ( ph  ->  A  e.  D )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  B  e.  V )   &    |-  ( ( ph  /\  x  =  A )  ->  (
 ( F `  A )  =  B  ->  ps ) )   &    |-  F/_ x F   &    |-  F/ x ps   =>    |-  ( ph  ->  ( F  =  ( x  e.  D  |->  B )  ->  ps ) )
 
Theoremfvmptdv 5287* Alternate deduction version of fvmpt 5277, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |-  ( ph  ->  A  e.  D )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  B  e.  V )   &    |-  ( ( ph  /\  x  =  A )  ->  (
 ( F `  A )  =  B  ->  ps ) )   =>    |-  ( ph  ->  ( F  =  ( x  e.  D  |->  B )  ->  ps ) )
 
Theoremfvmptdv2 5288* Alternate deduction version of fvmpt 5277, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |-  ( ph  ->  A  e.  D )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  B  e.  V )   &    |-  ( ( ph  /\  x  =  A )  ->  B  =  C )   =>    |-  ( ph  ->  ( F  =  ( x  e.  D  |->  B )  ->  ( F `  A )  =  C ) )
 
Theoremmpteqb 5289* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 5293. (Contributed by Mario Carneiro, 14-Nov-2014.)
 |-  ( A. x  e.  A  B  e.  V  ->  ( ( x  e.  A  |->  B )  =  ( x  e.  A  |->  C )  <->  A. x  e.  A  B  =  C )
 )
 
Theoremfvmptt 5290* Closed theorem form of fvmpt 5277. (Contributed by Scott Fenton, 21-Feb-2013.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  ( ( A. x ( x  =  A  ->  B  =  C ) 
 /\  F  =  ( x  e.  D  |->  B )  /\  ( A  e.  D  /\  C  e.  V ) )  ->  ( F `  A )  =  C )
 
Theoremfvmptf 5291* Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5276 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ x C   &    |-  ( x  =  A  ->  B  =  C )   &    |-  F  =  ( x  e.  D  |->  B )   =>    |-  ( ( A  e.  D  /\  C  e.  V )  ->  ( F `  A )  =  C )
 
Theoremfvopab6 5292* Value of a function given by ordered-pair class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  F  =  { <. x ,  y >.  |  (
 ph  /\  y  =  B ) }   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  ( x  =  A  ->  B  =  C )   =>    |-  ( ( A  e.  D  /\  C  e.  R  /\  ps )  ->  ( F `  A )  =  C )
 
Theoremeqfnfv 5293* Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
 |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G 
 <-> 
 A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
 
Theoremeqfnfv2 5294* Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28. (Contributed by NM, 3-Aug-1994.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( F  =  G 
 <->  ( A  =  B  /\  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) ) )
 
Theoremeqfnfv3 5295* Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( F  =  G 
 <->  ( B  C_  A  /\  A. x  e.  A  ( x  e.  B  /\  ( F `  x )  =  ( G `  x ) ) ) ) )
 
Theoremeqfnfvd 5296* Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  A )   &    |-  (
 ( ph  /\  x  e.  A )  ->  ( F `  x )  =  ( G `  x ) )   =>    |-  ( ph  ->  F  =  G )
 
Theoremeqfnfv2f 5297* Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). This version of eqfnfv 5293 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.)
 |-  F/_ x F   &    |-  F/_ x G   =>    |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
 
Theoremeqfunfv 5298* Equality of functions is determined by their values. (Contributed by Scott Fenton, 19-Jun-2011.)
 |-  ( ( Fun  F  /\  Fun  G )  ->  ( F  =  G  <->  ( dom  F  =  dom  G 
 /\  A. x  e.  dom  F ( F `  x )  =  ( G `  x ) ) ) )
 
Theoremfvreseq 5299* Equality of restricted functions is determined by their values. (Contributed by NM, 3-Aug-1994.)
 |-  ( ( ( F  Fn  A  /\  G  Fn  A )  /\  B  C_  A )  ->  (
 ( F  |`  B )  =  ( G  |`  B )  <->  A. x  e.  B  ( F `  x )  =  ( G `  x ) ) )
 
Theoremfndmdif 5300* Two ways to express the locus of differences between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  \  G )  =  { x  e.  A  |  ( F `  x )  =/=  ( G `  x ) } )
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