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Theorem List for Intuitionistic Logic Explorer - 5201-5300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfvmptdf 5201* Alternate deduction version of fvmpt 5192, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
 D   &     V   &     F `    &     F/_ F   &     F/   =>     F  D  |->
 
Theoremfvmptdv 5202* Alternate deduction version of fvmpt 5192, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
 D   &     V   &     F `    =>     F  D  |->
 
Theoremfvmptdv2 5203* Alternate deduction version of fvmpt 5192, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
 D   &     V   &     C   =>     F  D  |->  F `  C
 
Theoremmpteqb 5204* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 5208. (Contributed by Mario Carneiro, 14-Nov-2014.)
 V  |->  |->  C  C
 
Theoremfvmptt 5205* Closed theorem form of fvmpt 5192. (Contributed by Scott Fenton, 21-Feb-2013.) (Revised by Mario Carneiro, 11-Sep-2015.)
 C  F  D  |->  D  C  V  F `
  C
 
Theoremfvmptf 5206* Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5191 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
 F/_   &     F/_ C   &     C   &     F  D  |->    =>     D  C  V  F `  C
 
Theoremfvopab6 5207* 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  { <. ,  >.  |  }   &       &     C   =>     D  C  R  F `
  C
 
Theoremeqfnfv 5208* 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  G  Fn  F  G  F `  G `
 
Theoremeqfnfv2 5209* 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  G  Fn  F  G  F `  G `
 
Theoremeqfnfv3 5210* Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.)
 F  Fn  G  Fn  F  G  C_  F `  G `
 
Theoremeqfnfvd 5211* Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.)
 F  Fn    &     G  Fn    &     F `  G `    =>     F  G
 
Theoremeqfnfv2f 5212* 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 5208 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.)
 F/_ F   &     F/_ G   =>     F  Fn  G  Fn  F  G  F `  G `
 
Theoremeqfunfv 5213* 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  dom  F F `  G `
 
Theoremfvreseq 5214* Equality of restricted functions is determined by their values. (Contributed by NM, 3-Aug-1994.)
 F  Fn  G  Fn  C_  F  |`  G  |`  F `  G `
 
Theoremfndmdif 5215* Two ways to express the locus of differences between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 F  Fn  G  Fn  dom  F 
 \  G 
 {  |  F `  =/=  G `  }
 
Theoremfndmdifcom 5216 The difference set between two functions is commutative. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 F  Fn  G  Fn  dom  F 
 \  G 
 dom  G  \  F
 
Theoremfndmin 5217* Two ways to express the locus of equality between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 F  Fn  G  Fn  dom  F  i^i  G 
 {  |  F `  G `  }
 
Theoremfneqeql 5218 Two functions are equal iff their equalizer is the whole domain. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 F  Fn  G  Fn  F  G  dom  F  i^i  G
 
Theoremfneqeql2 5219 Two functions are equal iff their equalizer contains the whole domain. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 F  Fn  G  Fn  F  G  C_  dom  F  i^i  G
 
Theoremfnreseql 5220 Two functions are equal on a subset iff their equalizer contains that subset. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 F  Fn  G  Fn  X  C_  F  |`  X  G  |`  X  X  C_  dom  F  i^i  G
 
Theoremchfnrn 5221* 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  F `  ran  F  C_  U.
 
Theoremfunfvop 5222 Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 14-Oct-1996.)
 Fun  F  dom  F  <. ,  F `  >.  F
 
Theoremfunfvbrb 5223 Two ways to say that is in the domain of  F. (Contributed by Mario Carneiro, 1-May-2014.)
 Fun  F  dom  F  F F `
 
Theoremfvimacnvi 5224 A member of a preimage is a function value argument. (Contributed by NM, 4-May-2007.)
 Fun  F  `' F "  F `
 
Theoremfvimacnv 5225 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 4920 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.)
 Fun  F  dom  F  F `
  `' F "
 
Theoremfunimass3 5226 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 5225 would be the special case of being a singleton, but it works this way round too." (Contributed by Raph Levien, 20-Nov-2006.)
 Fun  F  C_  dom  F  F "  C_  C_  `' F "
 
Theoremfunimass5 5227* A subclass of a preimage in terms of function values. (Contributed by NM, 15-May-2007.)
 Fun  F  C_  dom  F  C_  `' F "  F `
 
Theoremfunconstss 5228* Two ways of specifying that a function is constant on a subdomain. (Contributed by NM, 8-Mar-2007.)
 Fun  F  C_  dom  F  F `  C_  `' F " { }
 
Theoremelpreima 5229 Membership in the preimage of a set under a function. (Contributed by Jeff Madsen, 2-Sep-2009.)
 F  Fn  `' F " C  F `  C
 
Theoremfniniseg 5230 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  C  `' F " { }  C  F `
  C
 
Theoremfncnvima2 5231* Inverse images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 F  Fn  `' F "  {  |  F `
  }
 
Theoremfniniseg2 5232* Inverse point images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 F  Fn  `' F " { } 
 {  |  F `  }
 
Theoremfnniniseg2 5233* Support sets of functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 F  Fn  `' F " _V  \  { }  {  |  F `  =/=  }
 
Theoremrexsupp 5234* Existential quantification restricted to a support. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 F  Fn  `' F " _V  \  { Z }  F `  =/=  Z
 
Theoremunpreima 5235 Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.)
 Fun  F  `' F "  u.  `' F "  u.  `' F "
 
Theoreminpreima 5236 Preimage of an intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jun-2016.)
 Fun  F  `' F " 
 i^i  `' F "  i^i  `' F "
 
Theoremdifpreima 5237 Preimage of a difference. (Contributed by Mario Carneiro, 14-Jun-2016.)
 Fun  F  `' F " 
 \  `' F "  \  `' F "
 
Theoremrespreima 5238 The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.)
 Fun  F  `' F  |` 
 "  `' F "  i^i
 
Theoremfimacnv 5239 The preimage of the codomain of a mapping is the mapping's domain. (Contributed by FL, 25-Jan-2007.)
 F : -->  `' F "
 
Theoremfnopfv 5240 Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 30-Sep-2004.)
 F  Fn  <. ,  F `  >.  F
 
Theoremfvelrn 5241 A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.)
 Fun  F  dom  F  F `  ran  F
 
Theoremfnfvelrn 5242 A function's value belongs to its range. (Contributed by NM, 15-Oct-1996.)
 F  Fn  F `  ran  F
 
Theoremffvelrn 5243 A function's value belongs to its codomain. (Contributed by NM, 12-Aug-1999.)
 F :
 -->  C  F `
  C
 
Theoremffvelrni 5244 A function's value belongs to its codomain. (Contributed by NM, 6-Apr-2005.)
 F : -->   =>     C  F `  C
 
Theoremffvelrnda 5245 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
 F : -->   =>     C  F `  C
 
Theoremffvelrnd 5246 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
 F : -->   &     C    =>     F `
  C
 
Theoremrexrn 5247* Restricted existential quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)
 F `    =>     F  Fn  ran 
 F
 
Theoremralrn 5248* Restricted universal quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)
 F `    =>     F  Fn  ran 
 F
 
Theoremelrnrexdm 5249* For any element in the range of a function there is an element in the domain of the function for which the function value is the element of the range. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
 Fun  F  Y  ran  F  dom  F  Y  F `
 
Theoremelrnrexdmb 5250* For any element in the range of a function there is an element in the domain of the function for which the function value is the element of the range. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
 Fun  F  Y  ran  F  dom  F  Y  F `
 
Theoremeldmrexrn 5251* For any element in the domain of a function there is an element in the range of the function which is the function value for the element of the domain. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
 Fun  F  Y  dom  F  ran  F  F `
  Y
 
Theoremralrnmpt 5252* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)
 F  |->    &       =>     V  ran  F
 
Theoremrexrnmpt 5253* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)
 F  |->    &       =>     V  ran  F
 
Theoremdff2 5254 Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.)
 F : -->  F  Fn  F  C_  X.
 
Theoremdff3im 5255* Property of a mapping. (Contributed by Jim Kingdon, 4-Jan-2019.)
 F : -->  F  C_  X.  F
 
Theoremdff4im 5256* Property of a mapping. (Contributed by Jim Kingdon, 4-Jan-2019.)
 F : -->  F  C_  X.  F
 
Theoremdffo3 5257* An onto mapping expressed in terms of function values. (Contributed by NM, 29-Oct-2006.)
 F : -onto->  F : -->  F `
 
Theoremdffo4 5258* Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)
 F : -onto->  F : -->  F
 
Theoremdffo5 5259* Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)
 F : -onto->  F : -->  F
 
Theoremfoelrn 5260* Property of a surjective function. (Contributed by Jeff Madsen, 4-Jan-2011.)
 F : -onto->  C  C  F `
 
Theoremfoco2 5261 If a composition of two functions is surjective, then the function on the left is surjective. (Contributed by Jeff Madsen, 16-Jun-2011.)
 F :
 --> C  G :
 -->  F  o.  G : -onto-> C  F : -onto-> C
 
Theoremfmpt 5262* Functionality of the mapping operation. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
 F  |->  C   =>     C  F :
 -->
 
Theoremf1ompt 5263* Express bijection for a mapping operation. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by Mario Carneiro, 4-Dec-2016.)
 F  |->  C   =>     F : -1-1-onto->  C  C
 
Theoremfmpti 5264* Functionality of the mapping operation. (Contributed by NM, 19-Mar-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
 F  |->  C   &     C    =>     F : -->
 
Theoremfmptd 5265* Domain and codomain of the mapping operation; deduction form. (Contributed by Mario Carneiro, 13-Jan-2013.)
 C   &     F  |->    =>     F :
 --> C
 
Theoremffnfv 5266* A function maps to a class to which all values belong. (Contributed by NM, 3-Dec-2003.)
 F : -->  F  Fn  F `
 
Theoremffnfvf 5267 A function maps to a class to which all values belong. This version of ffnfv 5266 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 28-Sep-2006.)
 F/_   &     F/_   &     F/_ F   =>     F : -->  F  Fn  F `
 
Theoremfnfvrnss 5268* An upper bound for range determined by function values. (Contributed by NM, 8-Oct-2004.)
 F  Fn  F `  ran  F  C_
 
Theoremrnmptss 5269* The range of an operation given by the "maps to" notation as a subset. (Contributed by Thierry Arnoux, 24-Sep-2017.)
 F  |->    =>     C  ran  F 
 C_  C
 
Theoremfmpt2d 5270* Domain and codomain of the mapping operation; deduction form. (Contributed by NM, 27-Dec-2014.)
 V   &     F  |->    &     F `  C   =>     F : --> C
 
Theoremffvresb 5271* A necessary and sufficient condition for a restricted function. (Contributed by Mario Carneiro, 14-Nov-2013.)
 Fun  F  F  |`  : -->  dom 
 F  F `
 
Theoremf1oresrab 5272* Build a bijection between restricted abstract builders, given a bijection between the base classes, deduction version. (Contributed by Thierry Arnoux, 17-Aug-2018.)
 F  |->  C   &     F :
 -1-1-onto->   &     C    =>     F  |`  {  |  } : {  |  } -1-1-onto-> {  |  }
 
Theoremfmptco 5273* Composition of two functions expressed as ordered-pair class abstractions. If  F has the equation ( x + 2 ) and  G the equation ( 3 * z ) then  G  o.  F has the equation ( 3 * ( x + 2 ) ) . (Contributed by FL, 21-Jun-2012.) (Revised by Mario Carneiro, 24-Jul-2014.)
 R    &     F  |->  R   &     G  |->  S   &     R  S  T   =>     G  o.  F  |->  T
 
Theoremfmptcof 5274* Version of fmptco 5273 where needn't be distinct from . (Contributed by NM, 27-Dec-2014.)
 R    &     F  |->  R   &     G  |->  S   &     R  S  T   =>     G  o.  F  |->  T
 
Theoremfmptcos 5275* Composition of two functions expressed as mapping abstractions. (Contributed by NM, 22-May-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
 R    &     F  |->  R   &     G  |->  S   =>     G  o.  F  |->  [_ R  ]_ S
 
Theoremfcompt 5276* Express composition of two functions as a maps-to applying both in sequence. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
 : D
 --> E  : C
 --> D  o.  C  |->  ` 
 `
 
Theoremfcoconst 5277 Composition with a constant function. (Contributed by Stefan O'Rear, 11-Mar-2015.)
 F  Fn  X  Y  X  F  o.  I  X.  { Y }  I  X.  { F `  Y }
 
Theoremfsn 5278 A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 10-Dec-2003.)
 _V   &     _V   =>     F : { } --> { }  F  { <. ,  >. }
 
Theoremfsng 5279 A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 26-Oct-2012.)
 C  D  F : { } --> { }  F  { <. ,  >. }
 
Theoremfsn2 5280 A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by NM, 19-May-2004.)
 _V   =>     F : { } -->  F `  F  { <. ,  F `  >. }
 
Theoremxpsng 5281 The cross product of two singletons. (Contributed by Mario Carneiro, 30-Apr-2015.)
 V  W  { }  X.  { }  { <. ,  >. }
 
Theoremxpsn 5282 The cross product of two singletons. (Contributed by NM, 4-Nov-2006.)
 _V   &     _V   =>     { }  X.  { }  { <. ,  >. }
 
Theoremdfmpt 5283 Alternate definition for the "maps to" notation df-mpt 3811 (although it requires that be a set). (Contributed by NM, 24-Aug-2010.) (Revised by Mario Carneiro, 30-Dec-2016.)
 _V   =>     |->  U_  { <. ,  >. }
 
Theoremfnasrn 5284 A function expressed as the range of another function. (Contributed by Mario Carneiro, 22-Jun-2013.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
 _V   =>     |->  ran  |->  <. ,  >.
 
Theoremdfmptg 5285 Alternate definition for the "maps to" notation df-mpt 3811 (which requires that be a set). (Contributed by Jim Kingdon, 9-Jan-2019.)
 V  |->  U_  { <. ,  >. }
 
Theoremfnasrng 5286 A function expressed as the range of another function. (Contributed by Jim Kingdon, 9-Jan-2019.)
 V  |->  ran  |->  <. ,  >.
 
Theoremressnop0 5287 If is not in  C, then the restriction of a singleton of  <. ,  >. to  C is null. (Contributed by Scott Fenton, 15-Apr-2011.)
 C  { <. ,  >. }  |`  C  (/)
 
Theoremfpr 5288 A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 _V   &     _V   &     C  _V   &     D  _V   =>     =/=  { <. ,  C >. ,  <. ,  D >. } : { ,  } --> { C ,  D }
 
Theoremfprg 5289 A function with a domain of two elements. (Contributed by FL, 2-Feb-2014.)
 E  F  C  G  D  H  =/=  { <. ,  C >. ,  <. ,  D >. } : { ,  } --> { C ,  D }
 
Theoremftpg 5290 A function with a domain of three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
 X  U  Y  V  Z  W  F  G  C  H  X  =/=  Y  X  =/=  Z  Y  =/=  Z  { <. X ,  >. , 
 <. Y ,  >. , 
 <. Z ,  C >. } : { X ,  Y ,  Z } --> { ,  ,  C }
 
Theoremftp 5291 A function with a domain of three elements. (Contributed by Stefan O'Rear, 17-Oct-2014.) (Proof shortened by Alexander van der Vekens, 23-Jan-2018.)
 _V   &     _V   &     C  _V   &     X  _V   &     Y  _V   &     Z  _V   &     =/=    &     =/=  C   &     =/=  C   =>    
 { <. ,  X >. ,  <. ,  Y >. ,  <. C ,  Z >. } : { ,  ,  C } --> { X ,  Y ,  Z }
 
Theoremfnressn 5292 A function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)
 F  Fn  F  |`  { }  { <. ,  F `  >. }
 
Theoremfressnfv 5293 The value of a function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)
 F  Fn  F  |`  { } : { } --> C  F `
  C
 
Theoremfvconst 5294 The value of a constant function. (Contributed by NM, 30-May-1999.)
 F :
 --> { }  C  F `  C
 
Theoremfmptsn 5295* Express a singleton function in maps-to notation. (Contributed by NM, 6-Jun-2006.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear, 28-Feb-2015.)
 V  W  { <. ,  >. }  { }  |->
 
Theoremfmptap 5296* Append an additional value to a function. (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
 _V   &     _V   &     R  u.  { }  S   &     C    =>     R  |->  C  u.  { <. ,  >. }  S  |->  C
 
Theoremfmptapd 5297* Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.)
 _V   &     _V   &     R  u.  { }  S   &     C    =>     R  |->  C  u.  { <. ,  >. }  S  |->  C
 
Theoremfmptpr 5298* Express a pair function in maps-to notation. (Contributed by Thierry Arnoux, 3-Jan-2017.)
 V   &     W   &     C  X   &     D  Y   &     E  C   &     E  D   =>     { <. ,  C >. , 
 <. ,  D >. }  { ,  }  |->  E
 
Theoremfvresi 5299 The value of a restricted identity function. (Contributed by NM, 19-May-2004.)
 _I  |` 
 `
 
Theoremfvunsng 5300 Remove an ordered pair not participating in a function value. (Contributed by Jim Kingdon, 7-Jan-2019.)
 D  V  =/=  D  u.  { <. ,  C >. } `  D  `  D
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