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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cnveqb 5001 | Equality theorem for converse. (Contributed by FL, 19-Sep-2011.) |
Theorem | cnveq0 5002 | A relation empty iff its converse is empty. (Contributed by FL, 19-Sep-2011.) |
Theorem | dfrel3 5003 | Alternate definition of relation. (Contributed by NM, 14-May-2008.) |
Theorem | dmresv 5004 | The domain of a universal restriction. (Contributed by NM, 14-May-2008.) |
Theorem | rnresv 5005 | The range of a universal restriction. (Contributed by NM, 14-May-2008.) |
Theorem | dfrn4 5006 | Range defined in terms of image. (Contributed by NM, 14-May-2008.) |
Theorem | csbrng 5007 | Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.) |
Theorem | rescnvcnv 5008 | 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 5009 | The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.) |
Theorem | imacnvcnv 5010 | The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.) |
Theorem | dmsnm 5011* | The domain of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.) |
Theorem | rnsnm 5012* | The range of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.) |
Theorem | dmsn0 5013 | The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.) |
Theorem | cnvsn0 5014 | The converse of the singleton of the empty set is empty. (Contributed by Mario Carneiro, 30-Aug-2015.) |
Theorem | dmsn0el 5015 | The domain of a singleton is empty if the singleton's argument contains the empty set. (Contributed by NM, 15-Dec-2008.) |
Theorem | relsn2m 5016* | A singleton is a relation iff it has an inhabited domain. (Contributed by Jim Kingdon, 16-Dec-2018.) |
Theorem | dmsnopg 5017 | 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 5018 | The domain of an unordered pair of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Theorem | dmsnop 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.) |
Theorem | dmprop 5020 | The domain of an unordered pair of ordered pairs. (Contributed by NM, 13-Sep-2011.) |
Theorem | dmtpop 5021 | The domain of an unordered triple of ordered pairs. (Contributed by NM, 14-Sep-2011.) |
Theorem | cnvcnvsn 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.) |
Theorem | dmsnsnsng 5023 | The domain of the singleton of the singleton of a singleton. (Contributed by Jim Kingdon, 16-Dec-2018.) |
Theorem | rnsnopg 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.) |
Theorem | rnpropg 5025 | The range of a pair of ordered pairs is the pair of second members. (Contributed by Thierry Arnoux, 3-Jan-2017.) |
Theorem | rnsnop 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.) |
Theorem | op1sta 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.) |
Theorem | cnvsn 5028 | Converse of a singleton of an ordered pair. (Contributed by NM, 11-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Theorem | op2ndb 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.) |
Theorem | op2nda 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.) |
Theorem | cnvsng 5031 | Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.) |
Theorem | opswapg 5032 | Swap the members of an ordered pair. (Contributed by Jim Kingdon, 16-Dec-2018.) |
Theorem | elxp4 5033 | Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 5034. (Contributed by NM, 17-Feb-2004.) |
Theorem | elxp5 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.) |
Theorem | cnvresima 5035 | An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.) |
Theorem | resdm2 5036 | A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.) |
Theorem | resdmres 5037 | Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.) |
Theorem | imadmres 5038 | The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.) |
Theorem | mptpreima 5039* | The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
Theorem | mptiniseg 5040* | Converse singleton image of a function defined by maps-to. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
Theorem | dmmpt 5041 | The domain of the mapping operation in general. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 22-Mar-2015.) |
Theorem | dmmptss 5042* | The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.) |
Theorem | dmmptg 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.) |
Theorem | relco 5044 | A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.) |
Theorem | dfco2 5045* | Alternate definition of a class composition, using only one bound variable. (Contributed by NM, 19-Dec-2008.) |
Theorem | dfco2a 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.) |
Theorem | coundi 5047 | Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | coundir 5048 | Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | cores 5049 | Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | resco 5050 | Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.) |
Theorem | imaco 5051 | Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.) |
Theorem | rnco 5052 | The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.) |
Theorem | rnco2 5053 | The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.) |
Theorem | dmco 5054 | The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.) |
Theorem | coiun 5055* | Composition with an indexed union. (Contributed by NM, 21-Dec-2008.) |
Theorem | cocnvcnv1 5056 | A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.) |
Theorem | cocnvcnv2 5057 | A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.) |
Theorem | cores2 5058 | Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.) |
Theorem | co02 5059 | Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.) |
Theorem | co01 5060 | Composition with the empty set. (Contributed by NM, 24-Apr-2004.) |
Theorem | coi1 5061 | Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.) |
Theorem | coi2 5062 | Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.) |
Theorem | coires1 5063 | Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.) |
Theorem | coass 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.) |
Theorem | relcnvtr 5065 | A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.) |
Theorem | relssdmrn 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.) |
Theorem | cnvssrndm 5067 | The converse is a subset of the cartesian product of range and domain. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Theorem | cossxp 5068 | Composition as a subset of the cross product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.) |
Theorem | cossxp2 5069 | The composition of two relations is a relation, with bounds on its domain and codomain. (Contributed by BJ, 10-Jul-2022.) |
Theorem | cocnvres 5070 | Restricting a relation and a converse relation when they are composed together (Contributed by BJ, 10-Jul-2022.) |
Theorem | cocnvss 5071 | Upper bound for the composed of a relation and an inverse relation. (Contributed by BJ, 10-Jul-2022.) |
Theorem | relrelss 5072 | Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.) |
Theorem | unielrel 5073 | The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.) |
Theorem | relfld 5074 | The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.) |
Theorem | relresfld 5075 | Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.) |
Theorem | relcoi2 5076 | Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.) |
Theorem | relcoi1 5077 | Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.) |
Theorem | unidmrn 5078 | The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.) |
Theorem | relcnvfld 5079 | if is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.) |
Theorem | dfdm2 5080 | Alternate definition of domain df-dm 4556 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.) |
Theorem | unixpm 5081* | The double class union of an inhabited cross product is the union of its members. (Contributed by Jim Kingdon, 18-Dec-2018.) |
Theorem | unixp0im 5082 | The union of an empty cross product is empty. (Contributed by Jim Kingdon, 18-Dec-2018.) |
Theorem | cnvexg 5083 | The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.) |
Theorem | cnvex 5084 | The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 19-Dec-2003.) |
Theorem | relcnvexb 5085 | A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.) |
Theorem | ressn 5086 | Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) |
Theorem | cnviinm 5087* | The converse of an intersection is the intersection of the converse. (Contributed by Jim Kingdon, 18-Dec-2018.) |
Theorem | cnvpom 5088* | The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.) |
Theorem | cnvsom 5089* | The converse of a strict order relation is a strict order relation. (Contributed by Jim Kingdon, 19-Dec-2018.) |
Theorem | coexg 5090 | The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.) |
Theorem | coex 5091 | The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.) |
Theorem | xpcom 5092* | Composition of two cross products. (Contributed by Jim Kingdon, 20-Dec-2018.) |
Syntax | cio 5093 | Extend class notation with Russell's definition description binder (inverted iota). |
Theorem | iotajust 5094* | Soundness justification theorem for df-iota 5095. (Contributed by Andrew Salmon, 29-Jun-2011.) |
Definition | df-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.) |
Theorem | dfiota2 5096* | Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.) |
Theorem | nfiota1 5097 | Bound-variable hypothesis builder for the class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Theorem | nfiotadw 5098* | Bound-variable hypothesis builder for the class. (Contributed by Jim Kingdon, 21-Dec-2018.) |
Theorem | nfiotaw 5099* | Bound-variable hypothesis builder for the class. (Contributed by NM, 23-Aug-2011.) |
Theorem | cbviota 5100 | Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.) |
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