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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.)

Theoremcnveq0 5002 A relation empty iff its converse is empty. (Contributed by FL, 19-Sep-2011.)

Theoremdfrel3 5003 Alternate definition of relation. (Contributed by NM, 14-May-2008.)

Theoremdmresv 5004 The domain of a universal restriction. (Contributed by NM, 14-May-2008.)

Theoremrnresv 5005 The range of a universal restriction. (Contributed by NM, 14-May-2008.)

Theoremdfrn4 5006 Range defined in terms of image. (Contributed by NM, 14-May-2008.)

Theoremcsbrng 5007 Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.)

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.)

Theoremcnvcnvres 5009 The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.)

Theoremimacnvcnv 5010 The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.)

Theoremdmsnm 5011* The domain of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.)

Theoremrnsnm 5012* The range of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.)

Theoremdmsn0 5013 The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.)

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.)

Theoremrelsn2m 5016* A singleton is a relation iff it has an inhabited domain. (Contributed by Jim Kingdon, 16-Dec-2018.)

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.)

Theoremdmpropg 5018 The domain of an unordered pair of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)

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.)

Theoremdmprop 5020 The domain of an unordered pair of ordered pairs. (Contributed by NM, 13-Sep-2011.)

Theoremdmtpop 5021 The domain of an unordered triple of ordered pairs. (Contributed by NM, 14-Sep-2011.)

Theoremcnvcnvsn 5022 Double converse of a singleton of an ordered pair. (Unlike cnvsn 5028, this does not need any sethood assumptions on and .) (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremdmsnsnsng 5023 The domain of the singleton of the singleton of a singleton. (Contributed by Jim Kingdon, 16-Dec-2018.)

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.)

Theoremrnpropg 5025 The range of a pair of ordered pairs is the pair of second members. (Contributed by Thierry Arnoux, 3-Jan-2017.)

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.)

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.)

Theoremcnvsn 5028 Converse of a singleton of an ordered pair. (Contributed by NM, 11-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

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.)

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.)

Theoremcnvsng 5031 Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.)

Theoremopswapg 5032 Swap the members of an ordered pair. (Contributed by Jim Kingdon, 16-Dec-2018.)

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.)

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.)

Theoremcnvresima 5035 An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.)

Theoremresdm2 5036 A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.)

Theoremresdmres 5037 Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.)

Theoremimadmres 5038 The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.)

Theoremmptpreima 5039* The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Theoremmptiniseg 5040* Converse singleton image of a function defined by maps-to. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Theoremdmmpt 5041 The domain of the mapping operation in general. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 22-Mar-2015.)

Theoremdmmptss 5042* The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.)

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.)

Theoremrelco 5044 A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.)

Theoremdfco2 5045* Alternate definition of a class composition, using only one bound variable. (Contributed by NM, 19-Dec-2008.)

Theoremdfco2a 5046* Generalization of dfco2 5045, where can have any value between and . (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcoundi 5047 Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcoundir 5048 Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcores 5049 Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremresco 5050 Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)

Theoremimaco 5051 Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.)

Theoremrnco 5052 The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.)

Theoremrnco2 5053 The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.)

Theoremdmco 5054 The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.)

Theoremcoiun 5055* Composition with an indexed union. (Contributed by NM, 21-Dec-2008.)

Theoremcocnvcnv1 5056 A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.)

Theoremcocnvcnv2 5057 A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.)

Theoremcores2 5058 Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.)

Theoremco02 5059 Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.)

Theoremco01 5060 Composition with the empty set. (Contributed by NM, 24-Apr-2004.)

Theoremcoi1 5061 Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)

Theoremcoi2 5062 Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)

Theoremcoires1 5063 Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.)

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.)

Theoremrelcnvtr 5065 A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.)

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.)

Theoremcnvssrndm 5067 The converse is a subset of the cartesian product of range and domain. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremcossxp 5068 Composition as a subset of the cross product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.)

Theoremcossxp2 5069 The composition of two relations is a relation, with bounds on its domain and codomain. (Contributed by BJ, 10-Jul-2022.)

Theoremcocnvres 5070 Restricting a relation and a converse relation when they are composed together (Contributed by BJ, 10-Jul-2022.)

Theoremcocnvss 5071 Upper bound for the composed of a relation and an inverse relation. (Contributed by BJ, 10-Jul-2022.)

Theoremrelrelss 5072 Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.)

Theoremunielrel 5073 The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.)

Theoremrelfld 5074 The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.)

Theoremrelresfld 5075 Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.)

Theoremrelcoi2 5076 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.)

Theoremrelcoi1 5077 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.)

Theoremunidmrn 5078 The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.)

Theoremrelcnvfld 5079 if is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.)

Theoremdfdm2 5080 Alternate definition of domain df-dm 4556 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.)

Theoremunixpm 5081* The double class union of an inhabited cross product is the union of its members. (Contributed by Jim Kingdon, 18-Dec-2018.)

Theoremunixp0im 5082 The union of an empty cross product is empty. (Contributed by Jim Kingdon, 18-Dec-2018.)

Theoremcnvexg 5083 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.)

Theoremcnvex 5084 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 19-Dec-2003.)

Theoremrelcnvexb 5085 A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.)

Theoremressn 5086 Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)

Theoremcnviinm 5087* The converse of an intersection is the intersection of the converse. (Contributed by Jim Kingdon, 18-Dec-2018.)

Theoremcnvpom 5088* The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.)

Theoremcnvsom 5089* The converse of a strict order relation is a strict order relation. (Contributed by Jim Kingdon, 19-Dec-2018.)

Theoremcoexg 5090 The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.)

Theoremcoex 5091 The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.)

Theoremxpcom 5092* Composition of two cross products. (Contributed by Jim Kingdon, 20-Dec-2018.)

2.6.7  Definite description binder (inverted iota)

Syntaxcio 5093 Extend class notation with Russell's definition description binder (inverted iota).

Theoremiotajust 5094* Soundness justification theorem for df-iota 5095. (Contributed by Andrew Salmon, 29-Jun-2011.)

Definitiondf-iota 5095* 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 5106); otherwise, it evaluates to the empty set (see iotanul 5110). Russell used the inverted iota symbol 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.)

Theoremdfiota2 5096* Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.)

Theoremnfiota1 5097 Bound-variable hypothesis builder for the class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremnfiotadw 5098* Bound-variable hypothesis builder for the class. (Contributed by Jim Kingdon, 21-Dec-2018.)

Theoremnfiotaw 5099* Bound-variable hypothesis builder for the class. (Contributed by NM, 23-Aug-2011.)

Theoremcbviota 5100 Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)

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