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Theorem List for Intuitionistic Logic Explorer - 5001-5100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcnveqb 5001 Equality theorem for converse. (Contributed by FL, 19-Sep-2011.)
 |-  ( ( Rel  A  /\  Rel  B )  ->  ( A  =  B  <->  `' A  =  `' B ) )
 
Theoremcnveq0 5002 A relation empty iff its converse is empty. (Contributed by FL, 19-Sep-2011.)
 |-  ( Rel  A  ->  ( A  =  (/)  <->  `' A  =  (/) ) )
 
Theoremdfrel3 5003 Alternate definition of relation. (Contributed by NM, 14-May-2008.)
 |-  ( Rel  R  <->  ( R  |`  _V )  =  R )
 
Theoremdmresv 5004 The domain of a universal restriction. (Contributed by NM, 14-May-2008.)
 |- 
 dom  ( A  |`  _V )  =  dom  A
 
Theoremrnresv 5005 The range of a universal restriction. (Contributed by NM, 14-May-2008.)
 |- 
 ran  ( A  |`  _V )  =  ran  A
 
Theoremdfrn4 5006 Range defined in terms of image. (Contributed by NM, 14-May-2008.)
 |- 
 ran  A  =  ( A " _V )
 
Theoremcsbrng 5007 Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_
 ran  B  =  ran  [_ A  /  x ]_ B )
 
Theoremrescnvcnv 5008 The restriction of the double converse of a class. (Contributed by NM, 8-Apr-2007.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( `' `' A  |`  B )  =  ( A  |`  B )
 
Theoremcnvcnvres 5009 The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.)
 |-  `' `' ( A  |`  B )  =  ( `' `' A  |`  B )
 
Theoremimacnvcnv 5010 The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
 |-  ( `' `' A " B )  =  ( A " B )
 
Theoremdmsnm 5011* The domain of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.)
 |-  ( A  e.  ( _V  X.  _V )  <->  E. x  x  e. 
 dom  { A } )
 
Theoremrnsnm 5012* The range of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.)
 |-  ( A  e.  ( _V  X.  _V )  <->  E. x  x  e. 
 ran  { A } )
 
Theoremdmsn0 5013 The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.)
 |- 
 dom  { (/) }  =  (/)
 
Theoremcnvsn0 5014 The converse of the singleton of the empty set is empty. (Contributed by Mario Carneiro, 30-Aug-2015.)
 |-  `' { (/) }  =  (/)
 
Theoremdmsn0el 5015 The domain of a singleton is empty if the singleton's argument contains the empty set. (Contributed by NM, 15-Dec-2008.)
 |-  ( (/)  e.  A  ->  dom  { A }  =  (/) )
 
Theoremrelsn2m 5016* A singleton is a relation iff it has an inhabited domain. (Contributed by Jim Kingdon, 16-Dec-2018.)
 |-  A  e.  _V   =>    |-  ( Rel  { A } 
 <-> 
 E. x  x  e. 
 dom  { A } )
 
Theoremdmsnopg 5017 The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |-  ( B  e.  V  ->  dom  { <. A ,  B >. }  =  { A } )
 
Theoremdmpropg 5018 The domain of an unordered pair of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( B  e.  V  /\  D  e.  W )  ->  dom  { <. A ,  B >. ,  <. C ,  D >. }  =  { A ,  C }
 )
 
Theoremdmsnop 5019 The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  B  e.  _V   =>    |-  dom  { <. A ,  B >. }  =  { A }
 
Theoremdmprop 5020 The domain of an unordered pair of ordered pairs. (Contributed by NM, 13-Sep-2011.)
 |-  B  e.  _V   &    |-  D  e.  _V   =>    |- 
 dom  { <. A ,  B >. ,  <. C ,  D >. }  =  { A ,  C }
 
Theoremdmtpop 5021 The domain of an unordered triple of ordered pairs. (Contributed by NM, 14-Sep-2011.)
 |-  B  e.  _V   &    |-  D  e.  _V   &    |-  F  e.  _V   =>    |-  dom  {
 <. A ,  B >. , 
 <. C ,  D >. , 
 <. E ,  F >. }  =  { A ,  C ,  E }
 
Theoremcnvcnvsn 5022 Double converse of a singleton of an ordered pair. (Unlike cnvsn 5028, this does not need any sethood assumptions on  A and  B.) (Contributed by Mario Carneiro, 26-Apr-2015.)
 |-  `' `' { <. A ,  B >. }  =  `' { <. B ,  A >. }
 
Theoremdmsnsnsng 5023 The domain of the singleton of the singleton of a singleton. (Contributed by Jim Kingdon, 16-Dec-2018.)
 |-  ( A  e.  _V  ->  dom  { { { A } } }  =  { A } )
 
Theoremrnsnopg 5024 The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( A  e.  V  ->  ran  { <. A ,  B >. }  =  { B } )
 
Theoremrnpropg 5025 The range of a pair of ordered pairs is the pair of second members. (Contributed by Thierry Arnoux, 3-Jan-2017.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ran  { <. A ,  C >. ,  <. B ,  D >. }  =  { C ,  D }
 )
 
Theoremrnsnop 5026 The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   =>    |-  ran  { <. A ,  B >. }  =  { B }
 
Theoremop1sta 5027 Extract the first member of an ordered pair. (See op2nda 5030 to extract the second member and op1stb 4406 for an alternate version.) (Contributed by Raph Levien, 4-Dec-2003.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 U. dom  { <. A ,  B >. }  =  A
 
Theoremcnvsn 5028 Converse of a singleton of an ordered pair. (Contributed by NM, 11-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  `' { <. A ,  B >. }  =  { <. B ,  A >. }
 
Theoremop2ndb 5029 Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4406 to extract the first member and op2nda 5030 for an alternate version.) (Contributed by NM, 25-Nov-2003.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 |^| |^| |^| `' { <. A ,  B >. }  =  B
 
Theoremop2nda 5030 Extract the second member of an ordered pair. (See op1sta 5027 to extract the first member and op2ndb 5029 for an alternate version.) (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 U. ran  { <. A ,  B >. }  =  B
 
Theoremcnvsng 5031 Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  `' { <. A ,  B >. }  =  { <. B ,  A >. } )
 
Theoremopswapg 5032 Swap the members of an ordered pair. (Contributed by Jim Kingdon, 16-Dec-2018.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  U. `' { <. A ,  B >. }  =  <. B ,  A >. )
 
Theoremelxp4 5033 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 5034. (Contributed by NM, 17-Feb-2004.)
 |-  ( A  e.  ( B  X.  C )  <->  ( A  =  <. U. dom  { A } ,  U. ran  { A } >.  /\  ( U. dom  { A }  e.  B  /\  U. ran  { A }  e.  C )
 ) )
 
Theoremelxp5 5034 Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 5033 when the double intersection does not create class existence problems (caused by int0 3792). (Contributed by NM, 1-Aug-2004.)
 |-  ( A  e.  ( B  X.  C )  <->  ( A  =  <. |^| |^| A ,  U. ran  { A } >.  /\  ( |^| |^| A  e.  B  /\  U. ran  { A }  e.  C ) ) )
 
Theoremcnvresima 5035 An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.)
 |-  ( `' ( F  |`  A ) " B )  =  ( ( `' F " B )  i^i  A )
 
Theoremresdm2 5036 A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.)
 |-  ( A  |`  dom  A )  =  `' `' A
 
Theoremresdmres 5037 Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
 |-  ( A  |`  dom  ( A  |`  B ) )  =  ( A  |`  B )
 
Theoremimadmres 5038 The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
 |-  ( A " dom  ( A  |`  B ) )  =  ( A
 " B )
 
Theoremmptpreima 5039* The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( `' F " C )  =  { x  e.  A  |  B  e.  C }
 
Theoremmptiniseg 5040* Converse singleton image of a function defined by maps-to. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( C  e.  V  ->  ( `' F " { C } )  =  { x  e.  A  |  B  =  C } )
 
Theoremdmmpt 5041 The domain of the mapping operation in general. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 22-Mar-2015.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  dom 
 F  =  { x  e.  A  |  B  e.  _V
 }
 
Theoremdmmptss 5042* The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  dom 
 F  C_  A
 
Theoremdmmptg 5043* The domain of the mapping operation is the stated domain, if the function value is always a set. (Contributed by Mario Carneiro, 9-Feb-2013.) (Revised by Mario Carneiro, 14-Sep-2013.)
 |-  ( A. x  e.  A  B  e.  V  ->  dom  ( x  e.  A  |->  B )  =  A )
 
Theoremrelco 5044 A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.)
 |- 
 Rel  ( A  o.  B )
 
Theoremdfco2 5045* Alternate definition of a class composition, using only one bound variable. (Contributed by NM, 19-Dec-2008.)
 |-  ( A  o.  B )  =  U_ x  e. 
 _V  ( ( `' B " { x } )  X.  ( A " { x }
 ) )
 
Theoremdfco2a 5046* Generalization of dfco2 5045, where  C can have any value between  dom  A  i^i  ran 
B and  _V. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( ( dom  A  i^i  ran  B )  C_  C  ->  ( A  o.  B )  =  U_ x  e.  C  ( ( `' B " { x } )  X.  ( A " { x }
 ) ) )
 
Theoremcoundi 5047 Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( A  o.  ( B  u.  C ) )  =  ( ( A  o.  B )  u.  ( A  o.  C ) )
 
Theoremcoundir 5048 Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( ( A  u.  B )  o.  C )  =  ( ( A  o.  C )  u.  ( B  o.  C ) )
 
Theoremcores 5049 Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( ran  B  C_  C  ->  ( ( A  |`  C )  o.  B )  =  ( A  o.  B ) )
 
Theoremresco 5050 Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)
 |-  ( ( A  o.  B )  |`  C )  =  ( A  o.  ( B  |`  C ) )
 
Theoremimaco 5051 Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.)
 |-  ( ( A  o.  B ) " C )  =  ( A " ( B " C ) )
 
Theoremrnco 5052 The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.)
 |- 
 ran  ( A  o.  B )  =  ran  ( A  |`  ran  B )
 
Theoremrnco2 5053 The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.)
 |- 
 ran  ( A  o.  B )  =  ( A " ran  B )
 
Theoremdmco 5054 The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.)
 |- 
 dom  ( A  o.  B )  =  ( `' B " dom  A )
 
Theoremcoiun 5055* Composition with an indexed union. (Contributed by NM, 21-Dec-2008.)
 |-  ( A  o.  U_ x  e.  C  B )  =  U_ x  e.  C  ( A  o.  B )
 
Theoremcocnvcnv1 5056 A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.)
 |-  ( `' `' A  o.  B )  =  ( A  o.  B )
 
Theoremcocnvcnv2 5057 A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.)
 |-  ( A  o.  `' `' B )  =  ( A  o.  B )
 
Theoremcores2 5058 Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.)
 |-  ( dom  A  C_  C  ->  ( A  o.  `' ( `' B  |`  C ) )  =  ( A  o.  B ) )
 
Theoremco02 5059 Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.)
 |-  ( A  o.  (/) )  =  (/)
 
Theoremco01 5060 Composition with the empty set. (Contributed by NM, 24-Apr-2004.)
 |-  ( (/)  o.  A )  =  (/)
 
Theoremcoi1 5061 Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)
 |-  ( Rel  A  ->  ( A  o.  _I  )  =  A )
 
Theoremcoi2 5062 Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)
 |-  ( Rel  A  ->  (  _I  o.  A )  =  A )
 
Theoremcoires1 5063 Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.)
 |-  ( A  o.  (  _I  |`  B ) )  =  ( A  |`  B )
 
Theoremcoass 5064 Associative law for class composition. Theorem 27 of [Suppes] p. 64. Also Exercise 21 of [Enderton] p. 53. Interestingly, this law holds for any classes whatsoever, not just functions or even relations. (Contributed by NM, 27-Jan-1997.)
 |-  ( ( A  o.  B )  o.  C )  =  ( A  o.  ( B  o.  C ) )
 
Theoremrelcnvtr 5065 A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.)
 |-  ( Rel  R  ->  ( ( R  o.  R )  C_  R  <->  ( `' R  o.  `' R )  C_  `' R ) )
 
Theoremrelssdmrn 5066 A relation is included in the cross product of its domain and range. Exercise 4.12(t) of [Mendelson] p. 235. (Contributed by NM, 3-Aug-1994.)
 |-  ( Rel  A  ->  A 
 C_  ( dom  A  X.  ran  A ) )
 
Theoremcnvssrndm 5067 The converse is a subset of the cartesian product of range and domain. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  `' A  C_  ( ran 
 A  X.  dom  A )
 
Theoremcossxp 5068 Composition as a subset of the cross product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  ( A  o.  B )  C_  ( dom  B  X.  ran  A )
 
Theoremcossxp2 5069 The composition of two relations is a relation, with bounds on its domain and codomain. (Contributed by BJ, 10-Jul-2022.)
 |-  ( ph  ->  R  C_  ( A  X.  B ) )   &    |-  ( ph  ->  S 
 C_  ( B  X.  C ) )   =>    |-  ( ph  ->  ( S  o.  R ) 
 C_  ( A  X.  C ) )
 
Theoremcocnvres 5070 Restricting a relation and a converse relation when they are composed together (Contributed by BJ, 10-Jul-2022.)
 |-  ( S  o.  `' R )  =  (
 ( S  |`  dom  R )  o.  `' ( R  |`  dom  S ) )
 
Theoremcocnvss 5071 Upper bound for the composed of a relation and an inverse relation. (Contributed by BJ, 10-Jul-2022.)
 |-  ( S  o.  `' R )  C_  ( ran  ( R  |`  dom  S )  X.  ran  ( S  |` 
 dom  R ) )
 
Theoremrelrelss 5072 Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.)
 |-  ( ( Rel  A  /\  Rel  dom  A )  <->  A 
 C_  ( ( _V 
 X.  _V )  X.  _V ) )
 
Theoremunielrel 5073 The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.)
 |-  ( ( Rel  R  /\  A  e.  R ) 
 ->  U. A  e.  U. R )
 
Theoremrelfld 5074 The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.)
 |-  ( Rel  R  ->  U.
 U. R  =  ( dom  R  u.  ran  R ) )
 
Theoremrelresfld 5075 Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.)
 |-  ( Rel  R  ->  ( R  |`  U. U. R )  =  R )
 
Theoremrelcoi2 5076 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.)
 |-  ( Rel  R  ->  ( (  _I  |`  U. U. R )  o.  R )  =  R )
 
Theoremrelcoi1 5077 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.)
 |-  ( Rel  R  ->  ( R  o.  (  _I  |`  U. U. R ) )  =  R )
 
Theoremunidmrn 5078 The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.)
 |- 
 U. U. `' A  =  ( dom  A  u.  ran  A )
 
Theoremrelcnvfld 5079 if  R is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.)
 |-  ( Rel  R  ->  U.
 U. R  =  U. U. `' R )
 
Theoremdfdm2 5080 Alternate definition of domain df-dm 4556 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.)
 |- 
 dom  A  =  U. U. ( `' A  o.  A )
 
Theoremunixpm 5081* The double class union of an inhabited cross product is the union of its members. (Contributed by Jim Kingdon, 18-Dec-2018.)
 |-  ( E. x  x  e.  ( A  X.  B )  ->  U. U. ( A  X.  B )  =  ( A  u.  B ) )
 
Theoremunixp0im 5082 The union of an empty cross product is empty. (Contributed by Jim Kingdon, 18-Dec-2018.)
 |-  ( ( A  X.  B )  =  (/)  ->  U. ( A  X.  B )  =  (/) )
 
Theoremcnvexg 5083 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.)
 |-  ( A  e.  V  ->  `' A  e.  _V )
 
Theoremcnvex 5084 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 19-Dec-2003.)
 |-  A  e.  _V   =>    |-  `' A  e.  _V
 
Theoremrelcnvexb 5085 A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.)
 |-  ( Rel  R  ->  ( R  e.  _V  <->  `' R  e.  _V ) )
 
Theoremressn 5086 Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  ( A  |`  { B } )  =  ( { B }  X.  ( A " { B }
 ) )
 
Theoremcnviinm 5087* The converse of an intersection is the intersection of the converse. (Contributed by Jim Kingdon, 18-Dec-2018.)
 |-  ( E. y  y  e.  A  ->  `' |^|_ x  e.  A  B  =  |^|_
 x  e.  A  `' B )
 
Theoremcnvpom 5088* The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.)
 |-  ( E. x  x  e.  A  ->  ( R  Po  A  <->  `' R  Po  A ) )
 
Theoremcnvsom 5089* The converse of a strict order relation is a strict order relation. (Contributed by Jim Kingdon, 19-Dec-2018.)
 |-  ( E. x  x  e.  A  ->  ( R  Or  A  <->  `' R  Or  A ) )
 
Theoremcoexg 5090 The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  o.  B )  e.  _V )
 
Theoremcoex 5091 The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  o.  B )  e.  _V
 
Theoremxpcom 5092* Composition of two cross products. (Contributed by Jim Kingdon, 20-Dec-2018.)
 |-  ( E. x  x  e.  B  ->  (
 ( B  X.  C )  o.  ( A  X.  B ) )  =  ( A  X.  C ) )
 
2.6.7  Definite description binder (inverted iota)
 
Syntaxcio 5093 Extend class notation with Russell's definition description binder (inverted iota).
 class  ( iota x ph )
 
Theoremiotajust 5094* Soundness justification theorem for df-iota 5095. (Contributed by Andrew Salmon, 29-Jun-2011.)
 |- 
 U. { y  |  { x  |  ph }  =  { y } }  =  U. { z  |  { x  |  ph }  =  { z } }
 
Definitiondf-iota 5095* Define Russell's definition description binder, which can be read as "the unique  x such that  ph," where  ph ordinarily contains  x as a free variable. Our definition is meaningful only when there is exactly one  x such that  ph is true (see iotaval 5106); otherwise, it evaluates to the empty set (see iotanul 5110). 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 5118 (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 x ph )  =  U. { y  |  { x  |  ph }  =  { y } }
 
Theoremdfiota2 5096* Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.)
 |-  ( iota x ph )  =  U. { y  |  A. x ( ph  <->  x  =  y ) }
 
Theoremnfiota1 5097 Bound-variable hypothesis builder for the  iota class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x ( iota x ph )
 
Theoremnfiotadw 5098* Bound-variable hypothesis builder for the  iota class. (Contributed by Jim Kingdon, 21-Dec-2018.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/_ x ( iota y ps ) )
 
Theoremnfiotaw 5099* Bound-variable hypothesis builder for the  iota class. (Contributed by NM, 23-Aug-2011.)
 |- 
 F/ x ph   =>    |-  F/_ x ( iota y ph )
 
Theoremcbviota 5100 Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  F/ y ph   &    |-  F/ x ps   =>    |-  ( iota x ph )  =  ( iota y ps )
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