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Theorem List for Intuitionistic Logic Explorer - 4801-4900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremressn 4801 Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |`  { }  { }  X.  " { }
 
Theoremcnviinm 4802* The converse of an intersection is the intersection of the converse. (Contributed by Jim Kingdon, 18-Dec-2018.)
 `' |^|_  |^|_  `'
 
Theoremcnvpom 4803* The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.)
 R  Po  `' R  Po
 
Theoremcnvsom 4804* The converse of a strict order relation is a strict order relation. (Contributed by Jim Kingdon, 19-Dec-2018.)
 R  Or  `' R  Or
 
Theoremcoexg 4805 The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.)
 V  W  o.  _V
 
Theoremcoex 4806 The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.)
 _V   &     _V   =>     o.  _V
 
Theoremxpcom 4807* Composition of two cross products. (Contributed by Jim Kingdon, 20-Dec-2018.)
 X.  C  o.  X.  X.  C
 
2.6.7  Definite description binder (inverted iota)
 
Syntaxcio 4808 Extend class notation with Russell's definition description binder (inverted iota).
 iota
 
Theoremiotajust 4809* Soundness justification theorem for df-iota 4810. (Contributed by Andrew Salmon, 29-Jun-2011.)

 U. {  |  {  |  }  { } }  U.
 {  |  {  |  }  { } }
 
Definitiondf-iota 4810* Define Russell's definition description binder, which can be read as "the unique such that ," where ordinarily contains as a free variable. Our definition is meaningful only when there is exactly one such that is true (see iotaval 4821); otherwise, it evaluates to the empty set (see iotanul 4825). Russell used the inverted iota symbol 
iota to represent the binder.

Sometimes proofs need to expand an iota-based definition. That is, given "X = the x for which ... x ... x ..." holds, the proof needs to get to "... X ... X ...". A general strategy to do this is to use iotacl 4833 (for unbounded iota). This can be easier than applying a version that applies an explicit substitution, because substituting an iota into its own property always has a bound variable clash which must be first renamed or else guarded with NF.

(Contributed by Andrew Salmon, 30-Jun-2011.)

 iota  U. {  |  {  |  }  { } }
 
Theoremdfiota2 4811* Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.)
 iota  U. {  |  }
 
Theoremnfiota1 4812 Bound-variable hypothesis builder for the  iota class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
 F/_ iota
 
Theoremnfiotadxy 4813* Deduction version of nfiotaxy 4814. (Contributed by Jim Kingdon, 21-Dec-2018.)

 F/   &     F/   =>     F/_ iota
 
Theoremnfiotaxy 4814* Bound-variable hypothesis builder for the  iota class. (Contributed by NM, 23-Aug-2011.)

 F/   =>     F/_ iota
 
Theoremcbviota 4815 Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
   &    
 F/   &     F/   =>     iota  iota
 
Theoremcbviotav 4816* Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
   =>     iota  iota
 
Theoremsb8iota 4817 Variable substitution in description binder. Compare sb8eu 1910. (Contributed by NM, 18-Mar-2013.)

 F/   =>     iota  iota
 
Theoremiotaeq 4818 Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
 iota  iota
 
Theoremiotabi 4819 Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
 iota  iota
 
Theoremuniabio 4820* Part of Theorem 8.17 in [Quine] p. 56. This theorem serves as a lemma for the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)
 U. {  | 
 }
 
Theoremiotaval 4821* Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)
 iota
 
Theoremiotauni 4822 Equivalence between two different forms of  iota. (Contributed by Andrew Salmon, 12-Jul-2011.)
 iota  U. {  |  }
 
Theoremiotaint 4823 Equivalence between two different forms of  iota. (Contributed by Mario Carneiro, 24-Dec-2016.)
 iota  |^| {  |  }
 
Theoremiota1 4824 Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
 iota
 
Theoremiotanul 4825 Theorem 8.22 in [Quine] p. 57. This theorem is the result if there isn't exactly one that satisfies . (Contributed by Andrew Salmon, 11-Jul-2011.)
 iota  (/)
 
Theoremeuiotaex 4826 Theorem 8.23 in [Quine] p. 58, with existential uniqueness condition added. This theorem proves the existence of the  iota class under our definition. (Contributed by Jim Kingdon, 21-Dec-2018.)
 iota  _V
 
Theoremiotass 4827* Value of iota based on a proposition which holds only for values which are subsets of a given class. (Contributed by Mario Carneiro and Jim Kingdon, 21-Dec-2018.)
 C_  iota  C_
 
Theoremiota4 4828 Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)
 [. iota  ].
 
Theoremiota4an 4829 Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
 [. iota  ].
 
Theoremiota5 4830* A method for computing iota. (Contributed by NM, 17-Sep-2013.)
 V    =>     V  iota
 
Theoremiotabidv 4831* Formula-building deduction rule for iota. (Contributed by NM, 20-Aug-2011.)
   =>     iota  iota
 
Theoremiotabii 4832 Formula-building deduction rule for iota. (Contributed by Mario Carneiro, 2-Oct-2015.)
   =>     iota  iota
 
Theoremiotacl 4833 Membership law for descriptions.

This can useful for expanding an unbounded iota-based definition (see df-iota 4810).

(Contributed by Andrew Salmon, 1-Aug-2011.)

 iota  {  | 
 }
 
Theoremiota2df 4834 A condition that allows us to represent "the unique element such that " with a class expression . (Contributed by NM, 30-Dec-2014.)
 V   &       &       &     F/   &     F/   &     F/_   =>     iota
 
Theoremiota2d 4835* A condition that allows us to represent "the unique element such that " with a class expression . (Contributed by NM, 30-Dec-2014.)
 V   &       &       =>     iota
 
Theoremiota2 4836* The unique element such that . (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
   =>     iota
 
Theoremsniota 4837 A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.)
 {  |  }  { iota }
 
Theoremcsbiotag 4838* Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.)
 V  [_  ]_ iota  iota [.  ].
 
2.6.8  Functions
 
Syntaxwfun 4839 Extend the definition of a wff to include the function predicate. (Read: is a function.)
 Fun
 
Syntaxwfn 4840 Extend the definition of a wff to include the function predicate with a domain. (Read: is a function on .)
 Fn
 
Syntaxwf 4841 Extend the definition of a wff to include the function predicate with domain and codomain. (Read: 
F maps into .)
 F : -->
 
Syntaxwf1 4842 Extend the definition of a wff to include one-to-one functions. (Read:  F maps one-to-one into .) The notation ("1-1" above the arrow) is from Definition 6.15(5) of [TakeutiZaring] p. 27.
 F : -1-1->
 
Syntaxwfo 4843 Extend the definition of a wff to include onto functions. (Read:  F maps onto .) The notation ("onto" below the arrow) is from Definition 6.15(4) of [TakeutiZaring] p. 27.
 F : -onto->
 
Syntaxwf1o 4844 Extend the definition of a wff to include one-to-one onto functions. (Read:  F maps one-to-one onto .) The notation ("1-1" above the arrow and "onto" below the arrow) is from Definition 6.15(6) of [TakeutiZaring] p. 27.
 F : -1-1-onto->
 
Syntaxcfv 4845 Extend the definition of a class to include the value of a function. (Read: The value of  F at , or " F of .")
 F `
 
Syntaxwiso 4846 Extend the definition of a wff to include the isomorphism property. (Read:  H is an  R,  S isomorphism of onto .)
 H  Isom  R ,  S  ,
 
Definitiondf-fun 4847 Define predicate that determines if some class is a function. Definition 10.1 of [Quine] p. 65. For example, the expression  Fun  _I is true (funi 4875). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 3809 with the maps-to notation (see df-mpt 3811). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 4848), a function with a given domain and codomain (df-f 4849), a one-to-one function (df-f1 4850), an onto function (df-fo 4851), or a one-to-one onto function (df-f1o 4852). For alternate definitions, see dffun2 4855, dffun4 4856, dffun6 4859, dffun7 4871, dffun8 4872, and dffun9 4873. (Contributed by NM, 1-Aug-1994.)
 Fun  Rel  o.  `'  C_  _I
 
Definitiondf-fn 4848 Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. (Contributed by NM, 1-Aug-1994.)
 Fn  Fun  dom
 
Definitiondf-f 4849 Define a function (mapping) with domain and codomain. Definition 6.15(3) of [TakeutiZaring] p. 27. (Contributed by NM, 1-Aug-1994.)
 F : -->  F  Fn 
 ran  F  C_
 
Definitiondf-f1 4850 Define a one-to-one function. Compare Definition 6.15(5) of [TakeutiZaring] p. 27. We use their notation ("1-1" above the arrow). (Contributed by NM, 1-Aug-1994.)
 F : -1-1->  F : -->  Fun  `' F
 
Definitiondf-fo 4851 Define an onto function. Definition 6.15(4) of [TakeutiZaring] p. 27. We use their notation ("onto" under the arrow). (Contributed by NM, 1-Aug-1994.)
 F : -onto->  F  Fn 
 ran  F
 
Definitiondf-f1o 4852 Define a one-to-one onto function. Compare Definition 6.15(6) of [TakeutiZaring] p. 27. We use their notation ("1-1" above the arrow and "onto" below the arrow). (Contributed by NM, 1-Aug-1994.)
 F : -1-1-onto->  F : -1-1->  F : -onto->
 
Definitiondf-fv 4853* Define the value of a function,  F ` , also known as function application. For example,  _I  `  (/)  (/). Typically, function  F is defined using maps-to notation (see df-mpt 3811), but this is not required. For example, F = {  <. 2 , 6  >.,  <. 3 , 9  >. } -> ( F  ` 3 ) = 9 . We will later define two-argument functions using ordered pairs as  F  F `  <. ,  >.. This particular definition is quite convenient: it can be applied to any class and evaluates to the empty set when it is not meaningful. The left apostrophe notation originated with Peano and was adopted in Definition *30.01 of [WhiteheadRussell] p. 235, Definition 10.11 of [Quine] p. 68, and Definition 6.11 of [TakeutiZaring] p. 26. It means the same thing as the more familiar  F notation for a function's value at , i.e. " F of ," but without context-dependent notational ambiguity. (Contributed by NM, 1-Aug-1994.) Revised to use  iota. (Revised by Scott Fenton, 6-Oct-2017.)
 F `  iota F
 
Definitiondf-isom 4854* Define the isomorphism predicate. We read this as " H is an  R,  S isomorphism of onto ." Normally,  R and  S are ordering relations on and respectively. Definition 6.28 of [TakeutiZaring] p. 32, whose notation is the same as ours except that  R and  S are subscripts. (Contributed by NM, 4-Mar-1997.)
 H  Isom  R ,  S  ,  H : -1-1-onto->  R  H `
  S H `
 
Theoremdffun2 4855* Alternate definition of a function. (Contributed by NM, 29-Dec-1996.)
 Fun  Rel
 
Theoremdffun4 4856* Alternate definition of a function. Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 29-Dec-1996.)
 Fun  Rel  <. ,  >. 
 <. ,  >.
 
Theoremdffun5r 4857* A way of proving a relation is a function, analogous to mo2r 1949. (Contributed by Jim Kingdon, 27-May-2020.)
 Rel  <. ,  >.  Fun
 
Theoremdffun6f 4858* Definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
 F/_   &     F/_   =>     Fun  Rel
 
Theoremdffun6 4859* Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.)
 Fun  F  Rel  F  F
 
Theoremfunmo 4860* A function has at most one value for each argument. (Contributed by NM, 24-May-1998.)
 Fun  F  F
 
Theoremdffun4f 4861* Definition of function like dffun4 4856 but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 17-Mar-2019.)
 F/_   &     F/_   &     F/_   =>     Fun  Rel  <. ,  >. 
 <. ,  >.
 
Theoremfunrel 4862 A function is a relation. (Contributed by NM, 1-Aug-1994.)
 Fun  Rel
 
Theoremfunss 4863 Subclass theorem for function predicate. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Mario Carneiro, 24-Jun-2014.)
 C_  Fun  Fun
 
Theoremfuneq 4864 Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)
 Fun  Fun
 
Theoremfuneqi 4865 Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
   =>     Fun  Fun
 
Theoremfuneqd 4866 Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)
   =>     Fun  Fun
 
Theoremnffun 4867 Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.)
 F/_ F   =>    
 F/ Fun  F
 
Theoremsbcfung 4868 Distribute proper substitution through the function predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
 V  [.  ]. Fun  F  Fun  [_  ]_ F
 
Theoremfuneu 4869* There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 Fun  F  F  F
 
Theoremfuneu2 4870* There is exactly one value of a function. (Contributed by NM, 3-Aug-1994.)
 Fun  F  <. ,  >.  F  <. ,  >.  F
 
Theoremdffun7 4871* Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. (Enderton's definition is ambiguous because "there is only one" could mean either "there is at most one" or "there is exactly one." However, dffun8 4872 shows that it doesn't matter which meaning we pick.) (Contributed by NM, 4-Nov-2002.)
 Fun  Rel  dom
 
Theoremdffun8 4872* Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 4871. (Contributed by NM, 4-Nov-2002.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 Fun  Rel  dom
 
Theoremdffun9 4873* Alternate definition of a function. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.)
 Fun  Rel  dom  ran
 
Theoremfunfn 4874 An equivalence for the function predicate. (Contributed by NM, 13-Aug-2004.)
 Fun  Fn  dom
 
Theoremfuni 4875 The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. (Contributed by NM, 30-Apr-1998.)

 Fun  _I
 
Theoremnfunv 4876 The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.)
 Fun  _V
 
Theoremfunopg 4877 A Kuratowski ordered pair is a function only if its components are equal. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
 V  W  Fun  <. ,  >.
 
Theoremfunopab 4878* A class of ordered pairs is a function when there is at most one second member for each pair. (Contributed by NM, 16-May-1995.)
 Fun  { <. ,  >.  |  }
 
Theoremfunopabeq 4879* A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.)

 Fun  { <. ,  >.  |  }
 
Theoremfunopab4 4880* A class of ordered pairs of values in the form used by df-mpt 3811 is a function. (Contributed by NM, 17-Feb-2013.)

 Fun  { <. ,  >.  |  }
 
Theoremfunmpt 4881 A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.)

 Fun  |->
 
Theoremfunmpt2 4882 Functionality of a class given by a "maps to" notation. (Contributed by FL, 17-Feb-2008.) (Revised by Mario Carneiro, 31-May-2014.)
 F  |->    =>     Fun 
 F
 
Theoremfunco 4883 The composition of two functions is a function. Exercise 29 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 Fun  F  Fun  G  Fun  F  o.  G
 
Theoremfunres 4884 A restriction of a function is a function. Compare Exercise 18 of [TakeutiZaring] p. 25. (Contributed by NM, 16-Aug-1994.)
 Fun  F  Fun  F  |`
 
Theoremfunssres 4885 The restriction of a function to the domain of a subclass equals the subclass. (Contributed by NM, 15-Aug-1994.)
 Fun  F  G  C_  F  F  |`  dom  G  G
 
Theoremfun2ssres 4886 Equality of restrictions of a function and a subclass. (Contributed by NM, 16-Aug-1994.)
 Fun  F  G  C_  F  C_  dom  G  F  |`  G  |`
 
Theoremfunun 4887 The union of functions with disjoint domains is a function. Theorem 4.6 of [Monk1] p. 43. (Contributed by NM, 12-Aug-1994.)
 Fun  F  Fun  G  dom  F  i^i  dom  G  (/)  Fun  F  u.  G
 
Theoremfuncnvsn 4888 The converse singleton of an ordered pair is a function. This is equivalent to funsn 4891 via cnvsn 4746, but stating it this way allows us to skip the sethood assumptions on and . (Contributed by NM, 30-Apr-2015.)

 Fun  `' { <. ,  >. }
 
Theoremfunsng 4889 A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 28-Jun-2011.)
 V  W  Fun  { <. ,  >. }
 
Theoremfnsng 4890 Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.)
 V  W  { <. ,  >. }  Fn  { }
 
Theoremfunsn 4891 A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 12-Aug-1994.)
 _V   &     _V   =>     Fun  { <. ,  >. }
 
Theoremfunprg 4892 A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.)
 V  W  C  X  D  Y  =/=  Fun  { <. ,  C >. ,  <. ,  D >. }
 
Theoremfuntpg 4893 A set of three pairs is a function if their first members are different. (Contributed by Alexander van der Vekens, 5-Dec-2017.)
 X  U  Y  V  Z  W  F  G  C  H  X  =/=  Y  X  =/=  Z  Y  =/=  Z  Fun  {
 <. X ,  >. , 
 <. Y ,  >. , 
 <. Z ,  C >. }
 
Theoremfunpr 4894 A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)
 _V   &     _V   &     C  _V   &     D  _V   =>     =/=  Fun  { <. ,  C >. ,  <. ,  D >. }
 
Theoremfuntp 4895 A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
 _V   &     _V   &     C  _V   &     D  _V   &     E  _V   &     F  _V   =>     =/=  =/=  C  =/=  C  Fun  { <. ,  D >. ,  <. ,  E >. ,  <. C ,  F >. }
 
Theoremfnsn 4896 Functionality and domain of the singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 _V   &     _V   =>     { <. ,  >. }  Fn  { }
 
Theoremfnprg 4897 Function with a domain of two different values. (Contributed by FL, 26-Jun-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
 V  W  C  X  D  Y  =/=  { <. ,  C >. ,  <. ,  D >. }  Fn  { ,  }
 
Theoremfntpg 4898 Function with a domain of three different values. (Contributed by Alexander van der Vekens, 5-Dec-2017.)
 X  U  Y  V  Z  W  F  G  C  H  X  =/=  Y  X  =/=  Z  Y  =/=  Z  { <. X ,  >. , 
 <. Y ,  >. , 
 <. Z ,  C >. }  Fn  { X ,  Y ,  Z }
 
Theoremfntp 4899 A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
 _V   &     _V   &     C  _V   &     D  _V   &     E  _V   &     F  _V   =>     =/=  =/=  C  =/=  C  { <. ,  D >. ,  <. ,  E >. ,  <. C ,  F >. }  Fn  { ,  ,  C }
 
Theoremfun0 4900 The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.)

 Fun  (/)
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