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Theorem List for Intuitionistic Logic Explorer - 5101-5200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremffoss 5101* Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145. (Contributed by NM, 10-May-1998.)
 F  _V   =>     F : -->  F : -onto->  C_
 
Theoremf11o 5102* Relationship between one-to-one and one-to-one onto function. (Contributed by NM, 4-Apr-1998.)
 F  _V   =>     F : -1-1->  F :
 -1-1-onto->  C_
 
Theoremf10 5103 The empty set maps one-to-one into any class. (Contributed by NM, 7-Apr-1998.)
 (/) : (/) -1-1->
 
Theoremf1o00 5104 One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.)
 F : (/) -1-1-onto->  F  (/)  (/)
 
Theoremfo00 5105 Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.)
 F : (/) -onto->  F  (/)  (/)
 
Theoremf1o0 5106 One-to-one onto mapping of the empty set. (Contributed by NM, 10-Sep-2004.)
 (/) : (/)
 -1-1-onto-> (/)
 
Theoremf1oi 5107 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  |`  : -1-1-onto->
 
Theoremf1ovi 5108 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 5109 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.)
 _V   &     _V   =>     { <. ,  >. } : { }
 -1-1-onto-> { }
 
Theoremf1osng 5110 A singleton of an ordered pair is one-to-one onto function. (Contributed by Mario Carneiro, 12-Jan-2013.)
 V  W  { <. ,  >. } : { } -1-1-onto-> { }
 
Theoremf1oprg 5111 An unordered pair of ordered pairs with different elements is a one-to-one onto function. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
 V  W  C  X  D  Y  =/=  C  =/=  D  { <. ,  >. , 
 <. C ,  D >. } : { ,  C } -1-1-onto-> { ,  D }
 
Theoremtz6.12-2 5112* 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.)
 F  F `  (/)
 
Theoremfveu 5113* The value of a function at a unique point. (Contributed by Scott Fenton, 6-Oct-2017.)
 F  F `
 
 U. {  |  F }
 
Theorembrprcneu 5114* If is a proper class, then there is no unique binary relationship with as the first element. (Contributed by Scott Fenton, 7-Oct-2017.)
 _V  F
 
Theoremfvprc 5115 A function's value at a proper class is the empty set. (Contributed by NM, 20-May-1998.)
 _V  F `  (/)
 
Theoremfv2 5116* 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 `  U. {  |  F  }
 
Theoremdffv3g 5117* A definition of function value in terms of iota. (Contributed by Jim Kingdon, 29-Dec-2018.)
 V  F `  iota  F
 " { }
 
Theoremdffv4g 5118* 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 4637), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.)
 V  F `  U. {  |  F " { }  { } }
 
Theoremelfv 5119* Membership in a function value. (Contributed by NM, 30-Apr-2004.)
 F `  F
 
Theoremfveq1 5120 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
 F  G  F `  G `
 
Theoremfveq2 5121 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
 F `  F `
 
Theoremfveq1i 5122 Equality inference for function value. (Contributed by NM, 2-Sep-2003.)
 F  G   =>     F `  G `
 
Theoremfveq1d 5123 Equality deduction for function value. (Contributed by NM, 2-Sep-2003.)
 F  G   =>     F `  G `
 
Theoremfveq2i 5124 Equality inference for function value. (Contributed by NM, 28-Jul-1999.)
   =>     F `  F `
 
Theoremfveq2d 5125 Equality deduction for function value. (Contributed by NM, 29-May-1999.)
   =>     F `  F `
 
Theoremfveq12i 5126 Equality deduction for function value. (Contributed by FL, 27-Jun-2014.)
 F  G   &       =>     F `  G `
 
Theoremfveq12d 5127 Equality deduction for function value. (Contributed by FL, 22-Dec-2008.)
 F  G   &       =>     F `  G `
 
Theoremnffv 5128 Bound-variable hypothesis builder for function value. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
 F/_ F   &     F/_   =>     F/_ F `
 
Theoremnffvmpt1 5129* Bound-variable hypothesis builder for mapping, special case. (Contributed by Mario Carneiro, 25-Dec-2016.)
 F/_  |-> 
 `  C
 
Theoremnffvd 5130 Deduction version of bound-variable hypothesis builder nffv 5128. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
 F/_ F   &     F/_   =>     F/_ F `
 
Theoremfunfveu 5131* A function has one value given an argument in its domain. (Contributed by Jim Kingdon, 29-Dec-2018.)
 Fun  F  dom  F  F
 
Theoremfvss 5132* The value of a function is a subset of if every element that could be a candidate for the value is a subset of . (Contributed by Mario Carneiro, 24-May-2019.)
 F  C_  F `  C_
 
Theoremfvssunirng 5133 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.)
 _V  F `  C_  U. ran  F
 
Theoremrelfvssunirn 5134 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 `  C_  U. ran  F
 
Theoremfunfvex 5135 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  dom  F  F `  _V
 
Theoremrelrnfvex 5136 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  _V  F `  _V
 
Theoremfvexg 5137 Evaluating a set function at a set exists. (Contributed by Mario Carneiro and Jim Kingdon, 28-May-2019.)
 F  V  W  F `  _V
 
Theoremfvex 5138 Evaluating a set function at a set exists. (Contributed by Mario Carneiro and Jim Kingdon, 28-May-2019.)
 F  V   &     W   =>     F `  _V
 
Theoremsefvex 5139 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  _V  F `  _V
 
Theoremfv3 5140* 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 `  {  |  F  F }
 
Theoremfvres 5141 The value of a restricted function. (Contributed by NM, 2-Aug-1994.)
 F  |` 
 `  F `
 
Theoremfunssfv 5142 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  dom  G  F `  G `
 
Theoremtz6.12-1 5143* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
 F  F  F `
 
Theoremtz6.12 5144* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 10-Jul-1994.)
 <. ,  >.  F  <. ,  >.  F  F `
 
Theoremtz6.12f 5145* Function value, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 30-Aug-1999.)
 F/_ F   =>     <. ,  >.  F  <. ,  >.  F  F `
 
Theoremtz6.12c 5146* Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
 F  F `  F
 
Theoremndmfvg 5147 The value of a class outside its domain is the empty set. (Contributed by Jim Kingdon, 15-Jan-2019.)
 _V  dom  F  F `
  (/)
 
Theoremrelelfvdm 5148 If a function value has a member, the argument belongs to the domain. (Contributed by Jim Kingdon, 22-Jan-2019.)
 Rel  F  F `
  dom  F
 
Theoremnfvres 5149 The value of a non-member of a restriction is the empty set. (Contributed by NM, 13-Nov-1995.)
 F  |`  `  (/)
 
Theoremnfunsn 5150 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  |`  { }  F `  (/)
 
Theorem0fv 5151 Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.)
 (/) `  (/)
 
Theoremcsbfv12g 5152 Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.)
 C  [_  ]_ F `  [_  ]_ F `  [_  ]_
 
Theoremcsbfv2g 5153* Move class substitution in and out of a function value. (Contributed by NM, 10-Nov-2005.)
 C  [_  ]_ F `  F ` 
 [_  ]_
 
Theoremcsbfvg 5154* Substitution for a function value. (Contributed by NM, 1-Jan-2006.)
 C  [_  ]_ F `  F `
 
Theoremfunbrfv 5155 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  F  F `
 
Theoremfunopfv 5156 The second element in an ordered pair member of a function is the function's value. (Contributed by NM, 19-Jul-1996.)
 Fun  F 
 <. ,  >.  F  F `
 
Theoremfnbrfvb 5157 Equivalence of function value and binary relation. (Contributed by NM, 19-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
 F  Fn  F `
  C  F C
 
Theoremfnopfvb 5158 Equivalence of function value and ordered pair membership. (Contributed by NM, 7-Nov-1995.)
 F  Fn  F `
  C  <. ,  C >.  F
 
Theoremfunbrfvb 5159 Equivalence of function value and binary relation. (Contributed by NM, 26-Mar-2006.)
 Fun  F  dom  F  F `
  F
 
Theoremfunopfvb 5160 Equivalence of function value and ordered pair membership. Theorem 4.3(ii) of [Monk1] p. 42. (Contributed by NM, 26-Jan-1997.)
 Fun  F  dom  F  F `
  <. ,  >.  F
 
Theoremfunbrfv2b 5161 Function value in terms of a binary relation. (Contributed by Mario Carneiro, 19-Mar-2014.)
 Fun  F  F  dom  F  F `
 
Theoremdffn5im 5162* 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 4881 and dmmptss 4760. (Contributed by Jim Kingdon, 31-Dec-2018.)
 F  Fn  F  |->  F `
 
Theoremfnrnfv 5163* 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  ran  F  {  |  F `  }
 
Theoremfvelrnb 5164* A member of a function's range is a value of the function. (Contributed by NM, 31-Oct-1995.)
 F  Fn  ran  F  F `
 
Theoremdfimafn 5165* Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.)
 Fun  F  C_  dom  F  F "  {  |  F `  }
 
Theoremdfimafn2 5166* 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  C_  dom  F  F "  U_  { F `  }
 
Theoremfunimass4 5167* Membership relation for the values of a function whose image is a subclass. (Contributed by Raph Levien, 20-Nov-2006.)
 Fun  F  C_  dom  F  F "  C_  F `
 
Theoremfvelima 5168* 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  F
 "  F `
 
Theoremfeqmptd 5169* Deduction form of dffn5im 5162. (Contributed by Mario Carneiro, 8-Jan-2015.)
 F : -->   =>     F  |->  F `
 
Theoremfeqresmpt 5170* Express a restricted function as a mapping. (Contributed by Mario Carneiro, 18-May-2016.)
 F : -->   &     C  C_    =>     F  |`  C  C  |->  F `
 
Theoremdffn5imf 5171* Representation of a function in terms of its values. (Contributed by Jim Kingdon, 31-Dec-2018.)
 F/_ F   =>     F  Fn  F  |->  F `
 
Theoremfvelimab 5172* 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  C_  C  F "  F `  C
 
Theoremfvi 5173 The value of the identity function. (Contributed by NM, 1-May-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
 V  _I  `
 
Theoremfniinfv 5174* The indexed intersection of a function's values is the intersection of its range. (Contributed by NM, 20-Oct-2005.)
 F  Fn  |^|_  F `  |^| ran  F
 
Theoremfnsnfv 5175 Singleton of function value. (Contributed by NM, 22-May-1998.)
 F  Fn  { F `
  }  F " { }
 
Theoremfnimapr 5176 The image of a pair under a function. (Contributed by Jeff Madsen, 6-Jan-2011.)
 F  Fn  C  F " { ,  C }  { F `  ,  F `  C }
 
Theoremssimaex 5177* The existence of a subimage. (Contributed by NM, 8-Apr-2007.)
 _V   =>     Fun  F  C_  F "  C_  F "
 
Theoremssimaexg 5178* The existence of a subimage. (Contributed by FL, 15-Apr-2007.)
 C  Fun  F  C_  F "  C_  F "
 
Theoremfunfvdm 5179 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  dom  F  F `  U. F " { }
 
Theoremfunfvdm2 5180* The value of a function. Definition of function value in [Enderton] p. 43. (Contributed by Jim Kingdon, 1-Jan-2019.)
 Fun  F  dom  F  F `  U. {  |  F }
 
Theoremfunfvdm2f 5181 The value of a function. Version of funfvdm2 5180 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 1-Jan-2019.)
 F/_   &     F/_ F   =>     Fun  F  dom  F  F `  U. {  |  F }
 
Theoremfvun1 5182 The value of a union when the argument is in the first domain. (Contributed by Scott Fenton, 29-Jun-2013.)
 F  Fn  G  Fn  i^i  (/)  X  F  u.  G `  X  F `  X
 
Theoremfvun2 5183 The value of a union when the argument is in the second domain. (Contributed by Scott Fenton, 29-Jun-2013.)
 F  Fn  G  Fn  i^i  (/)  X  F  u.  G `  X  G `  X
 
Theoremdmfco 5184 Domains of a function composition. (Contributed by NM, 27-Jan-1997.)
 Fun  G  dom  G  dom  F  o.  G  G `  dom  F
 
Theoremfvco2 5185 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  X  F  o.  G `  X  F `  G `  X
 
Theoremfvco 5186 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  dom  G  F  o.  G `  F `  G `
 
Theoremfvco3 5187 Value of a function composition. (Contributed by NM, 3-Jan-2004.) (Revised by Mario Carneiro, 26-Dec-2014.)
 G :
 -->  C  F  o.  G
 `  C  F `  G `  C
 
Theoremfvopab3g 5188* Value of a function given by ordered-pair class abstraction. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)
   &       &     C    &     F  { <. ,  >.  |  C  }   =>     C  D  F `
 
Theoremfvopab3ig 5189* Value of a function given by ordered-pair class abstraction. (Contributed by NM, 23-Oct-1999.)
   &       &     C    &     F  { <. ,  >.  |  C  }   =>     C  D  F `
 
Theoremfvmptss2 5190* 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.)
 D  C   &     F  |->    =>     F `  D  C_  C
 
Theoremfvmptg 5191* Value of a function given in maps-to notation. (Contributed by NM, 2-Oct-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
 C   &     F  D  |->    =>     D  C  R  F `  C
 
Theoremfvmpt 5192* Value of a function given in maps-to notation. (Contributed by NM, 17-Aug-2011.)
 C   &     F  D  |->    &     C  _V   =>     D  F `  C
 
Theoremfvmpts 5193* 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  C  |->    =>     C  [_  ]_  V  F `  [_  ]_
 
Theoremfvmpt3 5194* Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 C   &     F  D  |->    &     D  V   =>     D  F `  C
 
Theoremfvmpt3i 5195* Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Mario Carneiro, 11-Sep-2015.)
 C   &     F  D  |->    &     _V   =>     D  F `  C
 
Theoremfvmptd 5196* Deduction version of fvmpt 5192. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
 F  D  |->    &     C   &     D   &     C  V   =>     F `  C
 
Theoremfvmpt2 5197* Value of a function given by the "maps to" notation. (Contributed by FL, 21-Jun-2010.)
 F  |->    =>     C  F `
 
Theoremfvmptssdm 5198* 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  |->    =>     D  dom  F  C_  C  F `  D  C_  C
 
Theoremmptfvex 5199* Sufficient condition for a maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
 F  |->    =>     V  C  W  F `
  C 
 _V
 
Theoremfvmpt2d 5200* Deduction version of fvmpt2 5197. (Contributed by Thierry Arnoux, 8-Dec-2016.)
 F  |->    &     V   =>     F `
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