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Theorem List for Metamath Proof Explorer - 5701-5800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremf10 5701 The empty set maps one-to-one into any class. (Contributed by NM, 7-Apr-1998.)
 |-  (/) : (/) -1-1-> A
 
Theoremf1o00 5702 One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.)
 |-  ( F : (/) -1-1-onto-> A  <->  ( F  =  (/)  /\  A  =  (/) ) )
 
Theoremfo00 5703 Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.)
 |-  ( F : (/) -onto-> A  <-> 
 ( F  =  (/)  /\  A  =  (/) ) )
 
Theoremf1o0 5704 One-to-one onto mapping of the empty set. (Contributed by NM, 10-Sep-2004.)
 |-  (/) : (/)
 -1-1-onto-> (/)
 
Theoremf1oi 5705 A restriction of the identity relation is a one-to-one onto function. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  (  _I  |`  A ) : A -1-1-onto-> A
 
Theoremf1ovi 5706 The identity relation is a one-to-one onto function on the universe. (Contributed by NM, 16-May-2004.)
 |- 
 _I  : _V -1-1-onto-> _V
 
Theoremf1osn 5707 A singleton of an ordered pair is one-to-one onto function. (Contributed by NM, 18-May-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 { <. A ,  B >. } : { A }
 -1-1-onto-> { B }
 
Theoremf1osng 5708 A singleton of an ordered pair is one-to-one onto function. (Contributed by Mario Carneiro, 12-Jan-2013.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  { <. A ,  B >. } : { A } -1-1-onto-> { B } )
 
Theoremf1oprswap 5709 A two-element swap is a bijection on a pair. (Contributed by Mario Carneiro, 23-Jan-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  { <. A ,  B >. ,  <. B ,  A >. } : { A ,  B } -1-1-onto-> { A ,  B }
 )
 
Theoremf1oprg 5710 An unordered pair of ordered pairs with different elements is a one-to-one onto function, analogous to f1oprswap 5709. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
 |-  ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
 )  ->  ( ( A  =/=  C  /\  B  =/=  D )  ->  { <. A ,  B >. ,  <. C ,  D >. } : { A ,  C } -1-1-onto-> { B ,  D }
 ) )
 
Theoremtz6.12-2 5711* 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! x  A F x  ->  ( F `  A )  =  (/) )
 
Theoremfveu 5712* The value of a function at a unique point. (Contributed by Scott Fenton, 6-Oct-2017.)
 |-  ( E! x  A F x  ->  ( F `
  A )  = 
 U. { x  |  A F x } )
 
Theorembrprcneu 5713* If  A is a proper class, then there is no unique binary relationship with  A as the first element. (Contributed by Scott Fenton, 7-Oct-2017.)
 |-  ( -.  A  e.  _V 
 ->  -.  E! x  A F x )
 
Theoremfvprc 5714 A function's value at a proper class is the empty set. (Contributed by NM, 20-May-1998.)
 |-  ( -.  A  e.  _V 
 ->  ( F `  A )  =  (/) )
 
Theoremfv2 5715* 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 ) }
 
Theoremdffv3 5716* A definition of function value in terms of iota. (Contributed by Scott Fenton, 19-Feb-2013.)
 |-  ( F `  A )  =  ( iota x x  e.  ( F
 " { A }
 ) )
 
Theoremdffv4 5717* 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 5224), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.)
 |-  ( F `  A )  =  U. { x  |  ( F " { A } )  =  { x } }
 
Theoremelfv 5718* 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 5719 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
 |-  ( F  =  G  ->  ( F `  A )  =  ( G `  A ) )
 
Theoremfveq2 5720 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
 |-  ( A  =  B  ->  ( F `  A )  =  ( F `  B ) )
 
Theoremfveq1i 5721 Equality inference for function value. (Contributed by NM, 2-Sep-2003.)
 |-  F  =  G   =>    |-  ( F `  A )  =  ( G `  A )
 
Theoremfveq1d 5722 Equality deduction for function value. (Contributed by NM, 2-Sep-2003.)
 |-  ( ph  ->  F  =  G )   =>    |-  ( ph  ->  ( F `  A )  =  ( G `  A ) )
 
Theoremfveq2i 5723 Equality inference for function value. (Contributed by NM, 28-Jul-1999.)
 |-  A  =  B   =>    |-  ( F `  A )  =  ( F `  B )
 
Theoremfveq2d 5724 Equality deduction for function value. (Contributed by NM, 29-May-1999.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F `  A )  =  ( F `  B ) )
 
Theoremfveq12i 5725 Equality deduction for function value. (Contributed by FL, 27-Jun-2014.)
 |-  F  =  G   &    |-  A  =  B   =>    |-  ( F `  A )  =  ( G `  B )
 
Theoremfveq12d 5726 Equality deduction for function value. (Contributed by FL, 22-Dec-2008.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F `  A )  =  ( G `  B ) )
 
Theoremnffv 5727 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 5728* Bound-variable hypothesis builder for mapping, special case. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  F/_ x ( ( x  e.  A  |->  B ) `
  C )
 
Theoremnffvd 5729 Deduction version of bound-variable hypothesis builder nffv 5727. (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 ) )
 
Theoremcsbfv12g 5730 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 ) )
 
Theoremcsbfv12gALT 5731 Move class substitution in and out of a function value.(This is csbfv12g 5730 with a shortened proof, shortened by Alan Sare, 10-Nov-2012.) The proof is derived from the virtual deduction proof csbfv12gALTVD 28948. Although the proof is shorter, the total number of steps of all theorems used in the proof is probably longer. (Contributed by NM, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  C  -> 
 [_ A  /  x ]_ ( F `  B )  =  ( [_ A  /  x ]_ F ` 
 [_ A  /  x ]_ B ) )
 
Theoremcsbfv2g 5732* 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 5733* Substitution for a function value. (Contributed by NM, 1-Jan-2006.)
 |-  ( A  e.  C  -> 
 [_ A  /  x ]_ ( F `  x )  =  ( F `  A ) )
 
Theoremfvex 5734 The value of a class exists. Corollary 6.13 of [TakeutiZaring] p. 27. (Contributed by NM, 30-Dec-1996.)
 |-  ( F `  A )  e.  _V
 
Theoremfvif 5735 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 5736* 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 5737 The value of a restricted function. (Contributed by NM, 2-Aug-1994.)
 |-  ( A  e.  B  ->  ( ( F  |`  B ) `
  A )  =  ( F `  A ) )
 
Theoremfunssfv 5738 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 5739* 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 5740* 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 5741* 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 5742* 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 5743 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 5744 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 5745 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 5746 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 5747 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 5748 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 5749 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 5750 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 5751 If a function value is nonempty, its argument is a set. Deduction form of elfvex 5750. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  ( B `  C ) )   =>    |-  ( ph  ->  C  e.  _V )
 
Theoremnfvres 5752 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 5753 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 )  =  (/) )
 
Theoremfvfundmfvn0 5754 If a class' value at an argument is not the empty set, the argument is contained in the domain of the class, and the class restricted to the argument is a function. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( ( F `  A )  =/=  (/)  ->  ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) ) )
 
Theoremfv01 5755 Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.)
 |-  ( (/) `  A )  =  (/)
 
Theoremfveqres 5756 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 5757 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 5758 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 5759 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 5760 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 5761 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 5762 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 5763 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 5764* 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 5765* 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 5766* 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 5767* 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 5768* 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 5769* 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 5770* 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 5771* Deduction form of dffn5 5764. (Contributed by Mario Carneiro, 8-Jan-2015.)
 |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
 
Theoremfeqresmpt 5772* 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 5773* 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 5774* 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 5775 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 5776 The value of the identity function is a subset of the argument. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |-  (  _I  `  A )  C_  A
 
Theoremfniinfv 5777* 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 5778 Singleton of function value. (Contributed by NM, 22-May-1998.)
 |-  ( ( F  Fn  A  /\  B  e.  A )  ->  { ( F `
  B ) }  =  ( F " { B } ) )
 
Theoremfnimapr 5779 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 5780* 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 5781* 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 5782 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 5783* 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 5784 The value of a function. Version of funfv2 5783 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 5785 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 5786 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 5787 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 5788 Alternate definition of function value df-fv 5454 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 5789 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 5790 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 5791 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 5792 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 5793 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 5794* 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 5795* 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 5796* 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 5797* 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 5798* 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 5799* 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 5800* 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 )
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