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Theorem List for Metamath Proof Explorer - 5501-5600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfunbrfv2b 5501 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 5502* 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 5503* 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 5504* 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 5505* 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 5506* 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 5507* 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 5508* 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 5509* Deduction form of dffn5 5502. (Contributed by Mario Carneiro, 8-Jan-2015.)
 |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
 
Theoremfeqresmpt 5510* 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 5511* 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 5512* 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 5513 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 5514 The value of the identity function is a subset of the argument. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |-  (  _I  `  A )  C_  A
 
Theoremfniinfv 5515* 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 5516 Singleton of function value. (Contributed by NM, 22-May-1998.)
 |-  ( ( F  Fn  A  /\  B  e.  A )  ->  { ( F `
  B ) }  =  ( F " { B } ) )
 
Theoremfnimapr 5517 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 5518* 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 5519* 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 5520 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 5521* 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 5522 The value of a function. Version of funfv2 5521 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 5523 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 5524 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 5525 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 5526 Alternate definition of function value df-fv 4689 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 5527 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 5528 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 5529 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 5530 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 5531 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 5532* 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 5533* 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 5534* 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 5535* 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 5536* 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 5537* 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 5538* 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 5539* 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 5540* Deduction version of fvmpt 5536. (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 5541* 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 5542* 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 5543* 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 5544* 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 5472.) 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 5545* Alternate deduction version of fvmpt 5536, 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 5546* Alternate deduction version of fvmpt 5536, 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 5547* Alternate deduction version of fvmpt 5536, 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 5548* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 5556. (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 5549* Closed theorem form of fvmpt 5536. (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 5550* Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5534 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 5551* 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 5552 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 5552* 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 5534. (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 5553* 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 5554* 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 5555* 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 5556* 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 5557* 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 5558* 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 5559* 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 5560* 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 5556 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 5561* 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 5562* 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 5563* 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 5564 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 5565 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 5566* 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 5567 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 5568 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 5569 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 5570* 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 5571 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 5572 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 ) ) )
 
Theoremfvimacnvi 5573 A member of a preimage is a function value argument. (Contributed by NM, 4-May-2007.)
 |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  ->  ( F `  A )  e.  B )
 
Theoremfvimacnv 5574 The argument of a function value belongs to the preimage of any class containing the function value. Raph Levien remarks: "This proof is unsatisfying, because it seems to me that funimass2 5264 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.)
 |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  ( ( F `
  A )  e.  B  <->  A  e.  ( `' F " B ) ) )
 
Theoremfunimass3 5575 A kind of contraposition law that infers an image subclass from a subclass of a preimage. Raph Levien remarks: "Likely this could be proved directly, and fvimacnv 5574 would be the special case of  A being a singleton, but it works this way round too." (Contributed by Raph Levien, 20-Nov-2006.)
 |-  ( ( Fun  F  /\  A  C_  dom  F ) 
 ->  ( ( F " A )  C_  B  <->  A  C_  ( `' F " B ) ) )
 
Theoremfunimass5 5576* A subclass of a preimage in terms of function values. (Contributed by NM, 15-May-2007.)
 |-  ( ( Fun  F  /\  A  C_  dom  F ) 
 ->  ( A  C_  ( `' F " B )  <->  A. x  e.  A  ( F `  x )  e.  B ) )
 
Theoremfunconstss 5577* Two ways of specifying that a function is constant on a subdomain. (Contributed by NM, 8-Mar-2007.)
 |-  ( ( Fun  F  /\  A  C_  dom  F ) 
 ->  ( A. x  e.  A  ( F `  x )  =  B  <->  A 
 C_  ( `' F " { B } )
 ) )
 
TheoremfvimacnvALT 5578 Another proof of fvimacnv 5574, based on funimass3 5575. If funimass3 5575 is ever proved directly, as opposed to using funimacnv 5262 pointwise, then the proof of funimacnv 5262 should be replaced with this one. (Contributed by Raph Levien, 20-Nov-2006.) (Proof modification is discouraged.)
 |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  ( ( F `
  A )  e.  B  <->  A  e.  ( `' F " B ) ) )
 
Theoremelpreima 5579 Membership in the preimage of a set under a function. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( F  Fn  A  ->  ( B  e.  ( `' F " C )  <-> 
 ( B  e.  A  /\  ( F `  B )  e.  C )
 ) )
 
Theoremfniniseg 5580 Membership in the preimage of a singleton, under a function. (Contributed by Mario Carneiro, 12-May-2014.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
 |-  ( F  Fn  A  ->  ( C  e.  ( `' F " { B } )  <->  ( C  e.  A  /\  ( F `  C )  =  B ) ) )
 
Theoremfncnvima2 5581* Inverse images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( F  Fn  A  ->  ( `' F " B )  =  { x  e.  A  |  ( F `  x )  e.  B } )
 
Theoremfniniseg2 5582* Inverse point images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( F  Fn  A  ->  ( `' F " { B } )  =  { x  e.  A  |  ( F `  x )  =  B }
 )
 
Theoremfnniniseg2 5583* Support sets of functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( F  Fn  A  ->  ( `' F "
 ( _V  \  { B } ) )  =  { x  e.  A  |  ( F `  x )  =/=  B } )
 
Theoremrexsupp 5584* Existential quantification restricted to a support. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  ( F  Fn  A  ->  ( E. x  e.  ( `' F "
 ( _V  \  { Z } ) ) ph  <->  E. x  e.  A  (
 ( F `  x )  =/=  Z  /\  ph )
 ) )
 
Theoremunpreima 5585 Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( Fun  F  ->  ( `' F " ( A  u.  B ) )  =  ( ( `' F " A )  u.  ( `' F " B ) ) )
 
Theoreminpreima 5586 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 5587 Preimage of a difference. (Contributed by Mario Carneiro, 14-Jun-2016.)
 |-  ( Fun  F  ->  ( `' F " ( A 
 \  B ) )  =  ( ( `' F " A ) 
 \  ( `' F " B ) ) )
 
Theoremrespreima 5588 The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( Fun  F  ->  ( `' ( F  |`  B )
 " A )  =  ( ( `' F " A )  i^i  B ) )
 
Theoremiinpreima 5589* Preimage of an intersection. (Contributed by FL, 16-Apr-2012.)
 |-  ( ( Fun  F  /\  A  =/=  (/) )  ->  ( `' F " |^|_ x  e.  A  B )  = 
 |^|_ x  e.  A  ( `' F " B ) )
 
Theoremintpreima 5590* Preimage of an intersection. (Contributed by FL, 28-Apr-2012.)
 |-  ( ( Fun  F  /\  A  =/=  (/) )  ->  ( `' F " |^| A )  =  |^|_ x  e.  A  ( `' F " x ) )
 
Theoremfimacnv 5591 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 )
 
Theoremsuppss 5592* Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.)
 |-  ( ph  ->  F : A --> B )   &    |-  (
 ( ph  /\  k  e.  ( A  \  W ) )  ->  ( F `
  k )  =  Z )   =>    |-  ( ph  ->  ( `' F " ( _V  \  { Z } )
 )  C_  W )
 
Theoremsuppssr 5593 A function is zero outside its support. (Contributed by Mario Carneiro, 19-Dec-2014.)
 |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  ( `' F " ( _V  \  { Z } ) )  C_  W )   =>    |-  ( ( ph  /\  X  e.  ( A  \  W ) )  ->  ( F `
  X )  =  Z )
 
Theoremfnopfv 5594 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 5595 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 5596 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 5597 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 5598 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 5599 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 5600 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 )
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