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Theorem List for Metamath Proof Explorer - 5501-5600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfvif 5501 Move a conditional outside of a function. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( F `  if ( ph ,  A ,  B ) )  =  if ( ph ,  ( F `  A ) ,  ( F `  B ) )
 
Theoremfv3 5502* 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 5503 The value of a restricted function. (Contributed by NM, 2-Aug-1994.)
 |-  ( A  e.  B  ->  ( ( F  |`  B ) `
  A )  =  ( F `  A ) )
 
Theoremfunssfv 5504 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 5505* 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 5506* 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 5507* 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 )
 
Theoremtz6.12-2 5508* Function value when  F is not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
 |-  ( -.  E! y  A F y  ->  ( F `  A )  =  (/) )
 
Theoremtz6.12c 5509* 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 ) )
 
Theoremtz6.12i 5510 Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by Mario Carneiro, 17-Nov-2014.)
 |-  ( B  =/=  (/)  ->  (
 ( F `  A )  =  B  ->  A F B ) )
 
Theoremfvbr0 5511 Two possibilities for the behavior of a function value. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
 |-  ( X F ( F `  X )  \/  ( F `  X )  =  (/) )
 
Theoremfvrn0 5512 A function value is a member of the range plus null. (Contributed by Scott Fenton, 8-Jun-2011.) (Revised by Stefan O'Rear, 3-Jan-2015.)
 |-  ( F `  X )  e.  ( ran  F  u.  { (/) } )
 
Theoremfvssunirn 5513 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, 31-Aug-2015.)
 |-  ( F `  X )  C_  U. ran  F
 
Theoremndmfv 5514 The value of a class outside its domain is the empty set. (Contributed by NM, 24-Aug-1995.)
 |-  ( -.  A  e.  dom 
 F  ->  ( F `  A )  =  (/) )
 
Theoremndmfvrcl 5515 Reverse closure law for function with the empty set not in its domain. (Contributed by NM, 26-Apr-1996.)
 |- 
 dom  F  =  S   &    |-  -.  (/) 
 e.  S   =>    |-  ( ( F `  A )  e.  S  ->  A  e.  S )
 
Theoremelfvdm 5516 If a function value has a member, the argument belongs to the domain. (Contributed by NM, 12-Feb-2007.)
 |-  ( A  e.  ( F `  B )  ->  B  e.  dom  F )
 
Theoremelfvex 5517 If a function value has a member, the argument is a set. (Contributed by Mario Carneiro, 6-Nov-2015.)
 |-  ( A  e.  ( F `  B )  ->  B  e.  _V )
 
Theoremelfvexd 5518 If a function value is nonempty, its argument is a set. Deduction form of elfvex 5517. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  ( B `  C ) )   =>    |-  ( ph  ->  C  e.  _V )
 
Theoremnfvres 5519 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 5520 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 )  =  (/) )
 
Theoremfv01 5521 Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.)
 |-  ( (/) `  A )  =  (/)
 
Theoremfveqres 5522 Equal values imply equal values in a restriction. (Contributed by NM, 13-Nov-1995.)
 |-  ( ( F `  A )  =  ( G `  A )  ->  ( ( F  |`  B ) `
  A )  =  ( ( G  |`  B ) `
  A ) )
 
Theoremfunbrfv 5523 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 5524 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 5525 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 5526 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 5527 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 5528 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 5529 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 ) ) )
 
Theoremdffn5 5530* Representation of a function in terms of its values. (Contributed by FL, 14-Sep-2013.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
 |-  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
 
Theoremfnrnfv 5531* 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 5532* 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 5533* 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 5534* 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 5535* 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 5536* 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 5537* Deduction form of dffn5 5530. (Contributed by Mario Carneiro, 8-Jan-2015.)
 |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
 
Theoremfeqresmpt 5538* 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 ) ) )
 
Theoremdffn5f 5539* Representation of a function in terms of its values. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  F/_ x F   =>    |-  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
 
Theoremfvelimab 5540* 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 5541 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 )
 
Theoremfviss 5542 The value of the identity function is a subset of the argument. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |-  (  _I  `  A )  C_  A
 
Theoremfniinfv 5543* 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 5544 Singleton of function value. (Contributed by NM, 22-May-1998.)
 |-  ( ( F  Fn  A  /\  B  e.  A )  ->  { ( F `
  B ) }  =  ( F " { B } ) )
 
Theoremfnimapr 5545 The image of a pair under a funtion. (Contributed by Jeff Madsen, 6-Jan-2011.)
 |-  ( ( F  Fn  A  /\  B  e.  A  /\  C  e.  A ) 
 ->  ( F " { B ,  C }
 )  =  { ( F `  B ) ,  ( F `  C ) } )
 
Theoremssimaex 5546* 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 5547* 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 ) ) )
 
Theoremfunfv 5548 A simplified expression for the value of a function when we know it's a function. (Contributed by NM, 22-May-1998.)
 |-  ( Fun  F  ->  ( F `  A )  =  U. ( F
 " { A }
 ) )
 
Theoremfunfv2 5549* The value of a function. Definition of function value in [Enderton] p. 43. (Contributed by NM, 22-May-1998.)
 |-  ( Fun  F  ->  ( F `  A )  =  U. { y  |  A F y }
 )
 
Theoremfunfv2f 5550 The value of a function. Version of funfv2 5549 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 19-Feb-2006.)
 |-  F/_ y A   &    |-  F/_ y F   =>    |-  ( Fun  F  ->  ( F `  A )  =  U. { y  |  A F y }
 )
 
Theoremfvun 5551 Value of the union of two functions when the domains are separate. (Contributed by FL, 7-Nov-2011.)
 |-  ( ( ( Fun 
 F  /\  Fun  G ) 
 /\  (  dom  F  i^i  dom  G )  =  (/) )  ->  ( ( F  u.  G ) `
  A )  =  ( ( F `  A )  u.  ( G `  A ) ) )
 
Theoremfvun1 5552 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 5553 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 ) )
 
Theoremdffv2 5554 Alternate definition of function value df-fv 5229 that doesn't require dummy variables. (Contributed by NM, 4-Aug-2010.)
 |-  ( F `  A )  =  U. ( ( F " { A } )  \  U. U. ( ( ( F  |`  { A } )  o.  `' ( F  |`  { A } ) )  \  _I  ) )
 
Theoremdmfco 5555 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 5556 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 5557 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 5558 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 ) ) )
 
Theoremfvco4i 5559 Conditions for a composition to be expandable without conditions on the argument. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  (/)  =  ( F `  (/) )   &    |-  Fun  G   =>    |-  ( ( F  o.  G ) `  X )  =  ( F `  ( G `  X ) )
 
Theoremfvopab3g 5560* 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 5561* 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 ) )
 
Theoremfvmptg 5562* 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 )
 
Theoremfvmpti 5563* Value of a function given in maps-to notation. (Contributed by Mario Carneiro, 23-Apr-2014.)
 |-  ( x  =  A  ->  B  =  C )   &    |-  F  =  ( x  e.  D  |->  B )   =>    |-  ( A  e.  D  ->  ( F `  A )  =  (  _I  `  C ) )
 
Theoremfvmpt 5564* 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 5565* 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 5566* 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 5567* 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 5568* Deduction version of fvmpt 5564. (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 )
 
Theoremfvmpt2i 5569* Value of a function given by the "maps to" notation. (Contributed by Mario Carneiro, 23-Apr-2014.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( x  e.  A  ->  ( F `  x )  =  (  _I  `  B ) )
 
Theoremfvmpt2 5570* 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 )
 
Theoremfvmptss 5571* If all the values of the mapping are subsets of a class  C, then so is any evaluation of the mapping, even if  D is not in the base set  A. (Contributed by Mario Carneiro, 13-Feb-2015.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( A. x  e.  A  B  C_  C  ->  ( F `  D )  C_  C )
 
Theoremfvmptex 5572* Express a function  F whose value  B may not always be a set in terms of another function  G for which sethood is guaranteed. (Note that  (  _I  `  B ) is just shorthand for  if ( B  e.  _V ,  B ,  (/) ), and it is always a set by fvex 5500.) Note also that these functions are not the same; wherever  B
( C ) is not a set,  C is not in the domain of  F (so it evaluates to the empty set), but  C is in the domain of  G, and  G ( C ) is defined to be the empty set. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)
 |-  F  =  ( x  e.  A  |->  B )   &    |-  G  =  ( x  e.  A  |->  (  _I  `  B ) )   =>    |-  ( F `  C )  =  ( G `  C )
 
Theoremfvmptdf 5573* Alternate deduction version of fvmpt 5564, 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 5574* Alternate deduction version of fvmpt 5564, 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 5575* Alternate deduction version of fvmpt 5564, 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 5576* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 5584. (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 5577* Closed theorem form of fvmpt 5564. (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 5578* Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5562 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 )
 
Theoremfvmptnf 5579* The value of a function given by an ordered-pair class abstraction is the empty set when the class it would otherwise map to is a proper class. This version of fvmptn 5580 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  F/_ x A   &    |-  F/_ x C   &    |-  ( x  =  A  ->  B  =  C )   &    |-  F  =  ( x  e.  D  |->  B )   =>    |-  ( -.  C  e.  _V 
 ->  ( F `  A )  =  (/) )
 
Theoremfvmptn 5580* This somewhat non-intuitive theorem tells us the value of its function is the empty set when the class  C it would otherwise map to is a proper class. This is a technical lemma that can help eliminate redundant sethood antecedents otherwise required by fvmptg 5562. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 9-Sep-2013.)
 |-  ( x  =  D  ->  B  =  C )   &    |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( -.  C  e.  _V  ->  ( F `  D )  =  (/) )
 
Theoremfvmptss2 5581* 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.)
 |-  ( x  =  D  ->  B  =  C )   &    |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( F `  D )  C_  C
 
Theoremfvopab4ndm 5582* Value of a function given by an ordered-pair class abstraction, outside of its domain. (Contributed by NM, 28-Mar-2008.)
 |-  F  =  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }   =>    |-  ( -.  B  e.  A  ->  ( F `  B )  =  (/) )
 
Theoremfvopab6 5583* 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 5584* 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 5585* 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 5586* 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 5587* 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 5588* 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 5584 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 5589* 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 5590* 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 5591* 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 ) } )
 
Theoremfndmdifcom 5592 The difference set between two functions is commutative. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  (  F  \  G )  =  dom  (  G  \  F ) )
 
Theoremfndmdifeq0 5593 The difference set of two functions is empty if and only if the functions are equal. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  (  dom  (  F  \  G )  =  (/) 
 <->  F  =  G ) )
 
Theoremfndmin 5594* Two ways to express the locus of equality between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  (  F  i^i  G )  =  { x  e.  A  |  ( F `  x )  =  ( G `  x ) } )
 
Theoremfneqeql 5595 Two functions are equal iff their equalizer is the whole domain. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G 
 <-> 
 dom  (  F  i^i  G )  =  A ) )
 
Theoremfneqeql2 5596 Two functions are equal iff their equalizer contains the whole domain. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G 
 <->  A  C_  dom  (  F  i^i  G ) ) )
 
Theoremfnreseql 5597 Two functions are equal on a subset iff their equalizer contains that subset. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |-  ( ( F  Fn  A  /\  G  Fn  A  /\  X  C_  A )  ->  ( ( F  |`  X )  =  ( G  |`  X )  <->  X  C_  dom  (  F  i^i  G ) ) )
 
Theoremchfnrn 5598* The range of a choice function (a function that chooses an element from each member of its domain) is included in the union of its domain. (Contributed by NM, 31-Aug-1999.)
 |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  x )  ->  ran  F  C_  U. A )
 
Theoremfunfvop 5599 Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 14-Oct-1996.)
 |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  <. A ,  ( F `  A ) >.  e.  F )
 
Theoremfunfvbrb 5600 Two ways to say that  A is in the domain of  F. (Contributed by Mario Carneiro, 1-May-2014.)
 |-  ( Fun  F  ->  ( A  e.  dom  F  <->  A F ( F `  A ) ) )
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