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Theorem List for Intuitionistic Logic Explorer - 5301-5400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfvsn 5301 The value of a singleton of an ordered pair is the second member. (Contributed by NM, 12-Aug-1994.)
 _V   &     _V   =>     { <. ,  >. } `
 
Theoremfvsng 5302 The value of a singleton of an ordered pair is the second member. (Contributed by NM, 26-Oct-2012.)
 V  W  { <. ,  >. } `
 
Theoremfvsnun1 5303 The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 5304. (Contributed by NM, 23-Sep-2007.)
 _V   &     _V   &     G  { <. ,  >. }  u.  F  |`  C  \  { }   =>     G `
 
Theoremfvsnun2 5304 The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 5303. (Contributed by NM, 23-Sep-2007.)
 _V   &     _V   &     G  { <. ,  >. }  u.  F  |`  C  \  { }   =>     D  C  \  { }  G `  D  F `  D
 
Theoremfsnunf 5305 Adjoining a point to a function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
 F : S
 --> T  X  V  X  S  Y  T  F  u.  { <. X ,  Y >. } : S  u.  { X }
 --> T
 
Theoremfsnunfv 5306 Recover the added point from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by NM, 18-May-2017.)
 X  V  Y  W  X  dom  F  F  u.  { <. X ,  Y >. } `  X  Y
 
Theoremfsnunres 5307 Recover the original function from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
 F  Fn  S  X  S  F  u.  { <. X ,  Y >. }  |`  S  F
 
Theoremfvpr1 5308 The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)
 _V   &     C  _V   =>     =/=  { <. ,  C >. ,  <. ,  D >. } `  C
 
Theoremfvpr2 5309 The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)
 _V   &     D  _V   =>     =/=  { <. ,  C >. ,  <. ,  D >. } `  D
 
Theoremfvpr1g 5310 The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
 V  C  W  =/=  { <. ,  C >. ,  <. ,  D >. } `  C
 
Theoremfvpr2g 5311 The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
 V  D  W  =/=  { <. ,  C >. ,  <. ,  D >. } `  D
 
Theoremfvtp1g 5312 The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
 V  D  W  =/=  =/=  C  { <. ,  D >. ,  <. ,  E >. ,  <. C ,  F >. } `  D
 
Theoremfvtp2g 5313 The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
 V  E  W  =/=  =/=  C  { <. ,  D >. ,  <. ,  E >. ,  <. C ,  F >. } `  E
 
Theoremfvtp3g 5314 The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
 C  V  F  W  =/=  C  =/=  C  { <. ,  D >. ,  <. ,  E >. ,  <. C ,  F >. } `  C  F
 
Theoremfvtp1 5315 The first value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
 _V   &     D  _V   =>     =/=  =/=  C  { <. ,  D >. ,  <. ,  E >. ,  <. C ,  F >. } `  D
 
Theoremfvtp2 5316 The second value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
 _V   &     E  _V   =>     =/=  =/=  C  { <. ,  D >. ,  <. ,  E >. ,  <. C ,  F >. } `  E
 
Theoremfvtp3 5317 The third value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
 C  _V   &     F  _V   =>     =/=  C  =/=  C  { <. ,  D >. ,  <. ,  E >. ,  <. C ,  F >. } `  C  F
 
Theoremfvconst2g 5318 The value of a constant function. (Contributed by NM, 20-Aug-2005.)
 D  C 
 X.  { } `  C
 
Theoremfconst2g 5319 A constant function expressed as a cross product. (Contributed by NM, 27-Nov-2007.)
 C  F : --> { }  F  X.  { }
 
Theoremfvconst2 5320 The value of a constant function. (Contributed by NM, 16-Apr-2005.)
 _V   =>     C  X.  { } `  C
 
Theoremfconst2 5321 A constant function expressed as a cross product. (Contributed by NM, 20-Aug-1999.)
 _V   =>     F : --> { }  F  X.  { }
 
Theoremfconstfvm 5322* A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5321. (Contributed by Jim Kingdon, 8-Jan-2019.)
 F : --> { }  F  Fn  F `
 
Theoremfconst3m 5323* Two ways to express a constant function. (Contributed by Jim Kingdon, 8-Jan-2019.)
 F : --> { }  F  Fn  C_  `' F " { }
 
Theoremfconst4m 5324* Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)
 F : --> { }  F  Fn  `' F " { }
 
Theoremresfunexg 5325 The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. (Contributed by NM, 7-Apr-1995.) (Revised by Mario Carneiro, 22-Jun-2013.)
 Fun  C  |`  _V
 
Theoremfnex 5326 If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of resfunexg 5325. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 F  Fn  F  _V
 
Theoremfunex 5327 If the domain of a function exists, so the function. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of fnex 5326. (Note: Any resemblance between F.U.N.E.X. and "Have You Any Eggs" is purely a coincidence originated by Swedish chefs.) (Contributed by NM, 11-Nov-1995.)
 Fun  F  dom  F  F  _V
 
Theoremopabex 5328* Existence of a function expressed as class of ordered pairs. (Contributed by NM, 21-Jul-1996.)
 _V   &       =>     { <. ,  >.  |  }  _V
 
Theoremmptexg 5329* If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
 V  |->  _V
 
Theoremmptex 5330* If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by NM, 22-Apr-2005.) (Revised by Mario Carneiro, 20-Dec-2013.)
 _V   =>     |->  _V
 
Theoremfex 5331 If the domain of a mapping is a set, the function is a set. (Contributed by NM, 3-Oct-1999.)
 F :
 -->  C  F  _V
 
Theoremeufnfv 5332* A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.)
 _V   &     _V   =>     Fn 
 `
 
Theoremfunfvima 5333 A function's value in a preimage belongs to the image. (Contributed by NM, 23-Sep-2003.)
 Fun  F  dom  F  F `  F "
 
Theoremfunfvima2 5334 A function's value in an included preimage belongs to the image. (Contributed by NM, 3-Feb-1997.)
 Fun  F  C_  dom  F  F `  F "
 
Theoremfunfvima3 5335 A class including a function contains the function's value in the image of the singleton of the argument. (Contributed by NM, 23-Mar-2004.)
 Fun  F  F  C_  G  dom  F  F `  G " { }
 
Theoremfnfvima 5336 The function value of an operand in a set is contained in the image of that set, using the  Fn abbreviation. (Contributed by Stefan O'Rear, 10-Mar-2015.)
 F  Fn  S  C_  X  S  F `  X  F " S
 
Theoremrexima 5337* Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)
 F `    =>     F  Fn  C_  F "
 
Theoremralima 5338* Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)
 F `    =>     F  Fn  C_  F "
 
Theoremidref 5339* TODO: This is the same as issref 4650 (which has a much longer proof). Should we replace issref 4650 with this one? - NM 9-May-2016.

Two ways to state a relation is reflexive. (Adapted from Tarski.) (Contributed by FL, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) (Proof modification is discouraged.)

 _I  |`  C_  R  R
 
Theoremelabrex 5340* Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.)
 _V   =>     {  |  }
 
Theoremabrexco 5341* Composition of two image maps  C and . (Contributed by NM, 27-May-2013.)
 _V   &     C  D   =>     {  |  {  |  }  C }  {  |  D }
 
Theoremimaiun 5342* The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014.)
 " U_  C 
 U_  " C
 
Theoremimauni 5343* The image of a union is the indexed union of the images. Theorem 3K(a) of [Enderton] p. 50. (Contributed by NM, 9-Aug-2004.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)
 " U.  U_  "
 
Theoremfniunfv 5344* The indexed union of a function's values is the union of its range. Compare Definition 5.4 of [Monk1] p. 50. (Contributed by NM, 27-Sep-2004.)
 F  Fn  U_  F `  U. ran  F
 
Theoremfuniunfvdm 5345* The indexed union of a function's values is the union of its image under the index class. This theorem is a slight variation of fniunfv 5344. (Contributed by Jim Kingdon, 10-Jan-2019.)
 F  Fn  U_  F `  U. F
 "
 
Theoremfuniunfvdmf 5346* The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5345 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by Jim Kingdon, 10-Jan-2019.)
 F/_ F   =>     F  Fn  U_  F `  U. F
 "
 
Theoremeluniimadm 5347* Membership in the union of an image of a function. (Contributed by Jim Kingdon, 10-Jan-2019.)
 F  Fn  U. F "  F `
 
Theoremelunirn 5348* Membership in the union of the range of a function. (Contributed by NM, 24-Sep-2006.)
 Fun  F  U. ran  F  dom  F  F `
 
Theoremfnunirn 5349* Membership in a union of some function-defined family of sets. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 F  Fn  I  U. ran  F  I  F `
 
Theoremdff13 5350* A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 29-Oct-1996.)
 F : -1-1->  F : -->  F `  F `
 
Theoremf1veqaeq 5351 If the values of a one-to-one function for two arguments are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
 F :
 -1-1->  C  D  F `  C  F `  D  C  D
 
Theoremdff13f 5352* A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 31-Jul-2003.)
 F/_ F   &     F/_ F   =>     F : -1-1->  F : -->  F `  F `
 
Theoremf1mpt 5353* Express injection for a mapping operation. (Contributed by Mario Carneiro, 2-Jan-2017.)
 F  |->  C   &     C  D   =>     F : -1-1->  C  C  D
 
Theoremf1fveq 5354 Equality of function values for a one-to-one function. (Contributed by NM, 11-Feb-1997.)
 F :
 -1-1->  C  D  F `  C  F `  D  C  D
 
Theoremf1elima 5355 Membership in the image of a 1-1 map. (Contributed by Jeff Madsen, 2-Sep-2009.)
 F :
 -1-1->  X  Y  C_  F `
  X  F " Y  X  Y
 
Theoremf1imass 5356 Taking images under a one-to-one function preserves subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.)
 F :
 -1-1->  C 
 C_  D  C_  F " C  C_  F " D  C  C_  D
 
Theoremf1imaeq 5357 Taking images under a one-to-one function preserves equality. (Contributed by Stefan O'Rear, 30-Oct-2014.)
 F :
 -1-1->  C 
 C_  D  C_  F " C  F " D  C  D
 
Theoremf1imapss 5358 Taking images under a one-to-one function preserves proper subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.)
 F :
 -1-1->  C 
 C_  D  C_  F " C  C.  F " D  C  C.  D
 
Theoremdff1o6 5359* A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.)
 F : -1-1-onto->  F  Fn 
 ran  F  F `  F `
 
Theoremf1ocnvfv1 5360 The converse value of the value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)
 F :
 -1-1-onto->  C  `' F `  F `  C  C
 
Theoremf1ocnvfv2 5361 The value of the converse value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)
 F :
 -1-1-onto->  C  F `  `' F `  C  C
 
Theoremf1ocnvfv 5362 Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by Raph Levien, 10-Apr-2004.)
 F :
 -1-1-onto->  C  F `
  C  D  `' F `  D  C
 
Theoremf1ocnvfvb 5363 Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by NM, 20-May-2004.)
 F :
 -1-1-onto->  C  D  F `  C  D  `' F `  D  C
 
Theoremf1ocnvdm 5364 The value of the converse of a one-to-one onto function belongs to its domain. (Contributed by NM, 26-May-2006.)
 F :
 -1-1-onto->  C  `' F `  C
 
Theoremf1ocnvfvrneq 5365 If the values of a one-to-one function for two arguments from the range of the function are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
 F :
 -1-1->  C  ran  F  D  ran  F  `' F `  C  `' F `  D  C  D
 
Theoremfcof1 5366 An application is injective if a retraction exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 11-Nov-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
 F :
 -->  R  o.  F  _I  |`  F :
 -1-1->
 
Theoremfcofo 5367 An application is surjective if a section exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 17-Nov-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
 F :
 -->  S :
 -->  F  o.  S  _I  |`  F : -onto->
 
Theoremcbvfo 5368* Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
 F `    =>     F : -onto->
 
Theoremcbvexfo 5369* Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.)
 F `    =>     F : -onto->
 
Theoremcocan1 5370 An injection is left-cancelable. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.)
 F :
 -1-1-> C  H :
 -->  K :
 -->  F  o.  H  F  o.  K  H  K
 
Theoremcocan2 5371 A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
 F : -onto->  H  Fn  K  Fn  H  o.  F  K  o.  F  H  K
 
Theoremfcof1o 5372 Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.)
 F : -->  G : -->  F  o.  G  _I  |`  G  o.  F  _I  |`  F : -1-1-onto->  `' F  G
 
Theoremfoeqcnvco 5373 Condition for function equality in terms of vanishing of the composition with the converse. EDITORIAL: Is there a relation-algebraic proof of this? (Contributed by Stefan O'Rear, 12-Feb-2015.)
 F : -onto->  G : -onto->  F  G  F  o.  `' G  _I  |`
 
Theoremf1eqcocnv 5374 Condition for function equality in terms of vanishing of the composition with the inverse. (Contributed by Stefan O'Rear, 12-Feb-2015.)
 F :
 -1-1->  G :
 -1-1->  F  G  `' F  o.  G  _I  |`
 
Theoremfliftrel 5375*  F, a function lift, is a subset of  R  X.  S. (Contributed by Mario Carneiro, 23-Dec-2016.)
 F  ran  X  |->  <. ,  >.   &     X  R   &     X  S   =>     F  C_  R  X.  S
 
Theoremfliftel 5376* Elementhood in the relation  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
 F  ran  X  |->  <. ,  >.   &     X  R   &     X  S   =>     C F D  X  C  D
 
Theoremfliftel1 5377* Elementhood in the relation  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
 F  ran  X  |->  <. ,  >.   &     X  R   &     X  S   =>     X  F
 
Theoremfliftcnv 5378* Converse of the relation  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
 F  ran  X  |->  <. ,  >.   &     X  R   &     X  S   =>     `' F  ran  X  |->  <. ,  >.
 
Theoremfliftfun 5379* The function  F is the unique function defined by  F ` , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
 F  ran  X  |->  <. ,  >.   &     X  R   &     X  S   &     C   &     D   =>     Fun  F  X  X  C  D
 
Theoremfliftfund 5380* The function  F is the unique function defined by  F ` , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
 F  ran  X  |->  <. ,  >.   &     X  R   &     X  S   &     C   &     D   &     X  X  C  D   =>     Fun  F
 
Theoremfliftfuns 5381* The function  F is the unique function defined by  F ` , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
 F  ran  X  |->  <. ,  >.   &     X  R   &     X  S   =>     Fun  F  X  X  [_  ]_  [_  ]_  [_  ]_  [_  ]_
 
Theoremfliftf 5382* The domain and range of the function  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
 F  ran  X  |->  <. ,  >.   &     X  R   &     X  S   =>     Fun  F  F : ran  X  |->  --> S
 
Theoremfliftval 5383* The value of the function  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
 F  ran  X  |->  <. ,  >.   &     X  R   &     X  S   &     Y  C   &     Y  D   &     Fun  F   =>     Y  X  F `  C  D
 
Theoremisoeq1 5384 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
 H  G  H  Isom  R ,  S  ,  G  Isom  R ,  S  ,
 
Theoremisoeq2 5385 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
 R  T  H  Isom  R ,  S  ,  H  Isom  T ,  S  ,
 
Theoremisoeq3 5386 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
 S  T  H  Isom  R ,  S  ,  H  Isom  R ,  T  ,
 
Theoremisoeq4 5387 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
 C  H  Isom  R ,  S  ,  H  Isom  R ,  S  C ,
 
Theoremisoeq5 5388 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
 C  H  Isom  R ,  S  ,  H  Isom  R ,  S  ,  C
 
Theoremnfiso 5389 Bound-variable hypothesis builder for an isomorphism. (Contributed by NM, 17-May-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 F/_ H   &     F/_ R   &     F/_ S   &     F/_   &     F/_   =>     F/  H  Isom  R ,  S  ,
 
Theoremisof1o 5390 An isomorphism is a one-to-one onto function. (Contributed by NM, 27-Apr-2004.)
 H  Isom  R ,  S  ,  H : -1-1-onto->
 
Theoremisorel 5391 An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.)
 H  Isom  R ,  S  ,  C  D  C R D  H `
  C S H `  D
 
Theoremisoresbr 5392* A consequence of isomorphism on two relations for a function's restriction. (Contributed by Jim Kingdon, 11-Jan-2019.)
 F  |` 
 Isom  R ,  S  ,  F
 "  R  F `  S F `
 
Theoremisoid 5393 Identity law for isomorphism. Proposition 6.30(1) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)
 _I  |` 
 Isom  R ,  R  ,
 
Theoremisocnv 5394 Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)
 H  Isom  R ,  S  ,  `' H  Isom  S ,  R  ,
 
Theoremisocnv2 5395 Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.)
 H  Isom  R ,  S  ,  H  Isom  `' R ,  `' S ,
 
Theoremisores2 5396 An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)
 H  Isom  R ,  S  ,  H  Isom  R ,  S  i^i 
 X.  ,
 
Theoremisores1 5397 An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)
 H  Isom  R ,  S  ,  H  Isom  R  i^i  X.  ,  S ,
 
Theoremisores3 5398 Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.)
 H  Isom  R ,  S  ,  K  C_  X  H " K  H  |`  K  Isom  R ,  S  K ,  X
 
Theoremisotr 5399 Composition (transitive) law for isomorphism. Proposition 6.30(3) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
 H  Isom  R ,  S  ,  G  Isom  S ,  T  ,  C  G  o.  H  Isom  R ,  T  ,  C
 
Theoremisoini 5400 Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.)
 H  Isom  R ,  S  ,  D  H "  i^i  `' R " { D }  i^i  `' S " { H `  D }
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