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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | rescnvcnv 5001 | The restriction of the double converse of a class. (Contributed by NM, 8-Apr-2007.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | cnvcnvres 5002 | The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.) |
Theorem | imacnvcnv 5003 | The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.) |
Theorem | dmsnm 5004* | The domain of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.) |
Theorem | rnsnm 5005* | The range of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.) |
Theorem | dmsn0 5006 | The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.) |
Theorem | cnvsn0 5007 | The converse of the singleton of the empty set is empty. (Contributed by Mario Carneiro, 30-Aug-2015.) |
Theorem | dmsn0el 5008 | The domain of a singleton is empty if the singleton's argument contains the empty set. (Contributed by NM, 15-Dec-2008.) |
Theorem | relsn2m 5009* | A singleton is a relation iff it has an inhabited domain. (Contributed by Jim Kingdon, 16-Dec-2018.) |
Theorem | dmsnopg 5010 | The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Theorem | dmpropg 5011 | The domain of an unordered pair of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Theorem | dmsnop 5012 | 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.) |
Theorem | dmprop 5013 | The domain of an unordered pair of ordered pairs. (Contributed by NM, 13-Sep-2011.) |
Theorem | dmtpop 5014 | The domain of an unordered triple of ordered pairs. (Contributed by NM, 14-Sep-2011.) |
Theorem | cnvcnvsn 5015 | Double converse of a singleton of an ordered pair. (Unlike cnvsn 5021, this does not need any sethood assumptions on and .) (Contributed by Mario Carneiro, 26-Apr-2015.) |
Theorem | dmsnsnsng 5016 | The domain of the singleton of the singleton of a singleton. (Contributed by Jim Kingdon, 16-Dec-2018.) |
Theorem | rnsnopg 5017 | 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.) |
Theorem | rnpropg 5018 | The range of a pair of ordered pairs is the pair of second members. (Contributed by Thierry Arnoux, 3-Jan-2017.) |
Theorem | rnsnop 5019 | 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.) |
Theorem | op1sta 5020 | Extract the first member of an ordered pair. (See op2nda 5023 to extract the second member and op1stb 4399 for an alternate version.) (Contributed by Raph Levien, 4-Dec-2003.) |
Theorem | cnvsn 5021 | Converse of a singleton of an ordered pair. (Contributed by NM, 11-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Theorem | op2ndb 5022 | Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4399 to extract the first member and op2nda 5023 for an alternate version.) (Contributed by NM, 25-Nov-2003.) |
Theorem | op2nda 5023 | Extract the second member of an ordered pair. (See op1sta 5020 to extract the first member and op2ndb 5022 for an alternate version.) (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | cnvsng 5024 | Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.) |
Theorem | opswapg 5025 | Swap the members of an ordered pair. (Contributed by Jim Kingdon, 16-Dec-2018.) |
Theorem | elxp4 5026 | Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 5027. (Contributed by NM, 17-Feb-2004.) |
Theorem | elxp5 5027 | Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 5026 when the double intersection does not create class existence problems (caused by int0 3785). (Contributed by NM, 1-Aug-2004.) |
Theorem | cnvresima 5028 | An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.) |
Theorem | resdm2 5029 | A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.) |
Theorem | resdmres 5030 | Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.) |
Theorem | imadmres 5031 | The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.) |
Theorem | mptpreima 5032* | The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
Theorem | mptiniseg 5033* | Converse singleton image of a function defined by maps-to. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
Theorem | dmmpt 5034 | The domain of the mapping operation in general. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 22-Mar-2015.) |
Theorem | dmmptss 5035* | The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.) |
Theorem | dmmptg 5036* | 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.) |
Theorem | relco 5037 | A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.) |
Theorem | dfco2 5038* | Alternate definition of a class composition, using only one bound variable. (Contributed by NM, 19-Dec-2008.) |
Theorem | dfco2a 5039* | Generalization of dfco2 5038, where can have any value between and . (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | coundi 5040 | Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | coundir 5041 | Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | cores 5042 | Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | resco 5043 | Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.) |
Theorem | imaco 5044 | Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.) |
Theorem | rnco 5045 | The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.) |
Theorem | rnco2 5046 | The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.) |
Theorem | dmco 5047 | The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.) |
Theorem | coiun 5048* | Composition with an indexed union. (Contributed by NM, 21-Dec-2008.) |
Theorem | cocnvcnv1 5049 | A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.) |
Theorem | cocnvcnv2 5050 | A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.) |
Theorem | cores2 5051 | Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.) |
Theorem | co02 5052 | Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.) |
Theorem | co01 5053 | Composition with the empty set. (Contributed by NM, 24-Apr-2004.) |
Theorem | coi1 5054 | Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.) |
Theorem | coi2 5055 | Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.) |
Theorem | coires1 5056 | Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.) |
Theorem | coass 5057 | 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.) |
Theorem | relcnvtr 5058 | A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.) |
Theorem | relssdmrn 5059 | 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.) |
Theorem | cnvssrndm 5060 | The converse is a subset of the cartesian product of range and domain. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Theorem | cossxp 5061 | Composition as a subset of the cross product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.) |
Theorem | cossxp2 5062 | The composition of two relations is a relation, with bounds on its domain and codomain. (Contributed by BJ, 10-Jul-2022.) |
Theorem | cocnvres 5063 | Restricting a relation and a converse relation when they are composed together (Contributed by BJ, 10-Jul-2022.) |
Theorem | cocnvss 5064 | Upper bound for the composed of a relation and an inverse relation. (Contributed by BJ, 10-Jul-2022.) |
Theorem | relrelss 5065 | Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.) |
Theorem | unielrel 5066 | The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.) |
Theorem | relfld 5067 | The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.) |
Theorem | relresfld 5068 | Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.) |
Theorem | relcoi2 5069 | Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.) |
Theorem | relcoi1 5070 | Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.) |
Theorem | unidmrn 5071 | The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.) |
Theorem | relcnvfld 5072 | if is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.) |
Theorem | dfdm2 5073 | Alternate definition of domain df-dm 4549 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.) |
Theorem | unixpm 5074* | The double class union of an inhabited cross product is the union of its members. (Contributed by Jim Kingdon, 18-Dec-2018.) |
Theorem | unixp0im 5075 | The union of an empty cross product is empty. (Contributed by Jim Kingdon, 18-Dec-2018.) |
Theorem | cnvexg 5076 | The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.) |
Theorem | cnvex 5077 | The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 19-Dec-2003.) |
Theorem | relcnvexb 5078 | A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.) |
Theorem | ressn 5079 | Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) |
Theorem | cnviinm 5080* | The converse of an intersection is the intersection of the converse. (Contributed by Jim Kingdon, 18-Dec-2018.) |
Theorem | cnvpom 5081* | The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.) |
Theorem | cnvsom 5082* | The converse of a strict order relation is a strict order relation. (Contributed by Jim Kingdon, 19-Dec-2018.) |
Theorem | coexg 5083 | The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.) |
Theorem | coex 5084 | The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.) |
Theorem | xpcom 5085* | Composition of two cross products. (Contributed by Jim Kingdon, 20-Dec-2018.) |
Syntax | cio 5086 | Extend class notation with Russell's definition description binder (inverted iota). |
Theorem | iotajust 5087* | Soundness justification theorem for df-iota 5088. (Contributed by Andrew Salmon, 29-Jun-2011.) |
Definition | df-iota 5088* |
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 5099);
otherwise, it evaluates to the empty set (see iotanul 5103). 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 5111 (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.) |
Theorem | dfiota2 5089* | Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.) |
Theorem | nfiota1 5090 | Bound-variable hypothesis builder for the class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Theorem | nfiotadw 5091* | Bound-variable hypothesis builder for the class. (Contributed by Jim Kingdon, 21-Dec-2018.) |
Theorem | nfiotaw 5092* | Bound-variable hypothesis builder for the class. (Contributed by NM, 23-Aug-2011.) |
Theorem | cbviota 5093 | Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.) |
Theorem | cbviotav 5094* | Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.) |
Theorem | sb8iota 5095 | Variable substitution in description binder. Compare sb8eu 2012. (Contributed by NM, 18-Mar-2013.) |
Theorem | iotaeq 5096 | Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.) |
Theorem | iotabi 5097 | Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.) |
Theorem | uniabio 5098* | 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.) |
Theorem | iotaval 5099* | Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.) |
Theorem | iotauni 5100 | Equivalence between two different forms of . (Contributed by Andrew Salmon, 12-Jul-2011.) |
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