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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | dfrel3 4901 | Alternate definition of relation. (Contributed by NM, 14-May-2008.) |
Theorem | dmresv 4902 | The domain of a universal restriction. (Contributed by NM, 14-May-2008.) |
Theorem | rnresv 4903 | The range of a universal restriction. (Contributed by NM, 14-May-2008.) |
Theorem | dfrn4 4904 | Range defined in terms of image. (Contributed by NM, 14-May-2008.) |
Theorem | csbrng 4905 | Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.) |
Theorem | rescnvcnv 4906 | 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 4907 | The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.) |
Theorem | imacnvcnv 4908 | The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.) |
Theorem | dmsnm 4909* | The domain of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.) |
Theorem | rnsnm 4910* | The range of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.) |
Theorem | dmsn0 4911 | The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.) |
Theorem | cnvsn0 4912 | The converse of the singleton of the empty set is empty. (Contributed by Mario Carneiro, 30-Aug-2015.) |
Theorem | dmsn0el 4913 | The domain of a singleton is empty if the singleton's argument contains the empty set. (Contributed by NM, 15-Dec-2008.) |
Theorem | relsn2m 4914* | A singleton is a relation iff it has an inhabited domain. (Contributed by Jim Kingdon, 16-Dec-2018.) |
Theorem | dmsnopg 4915 | 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 4916 | The domain of an unordered pair of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Theorem | dmsnop 4917 | 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 4918 | The domain of an unordered pair of ordered pairs. (Contributed by NM, 13-Sep-2011.) |
Theorem | dmtpop 4919 | The domain of an unordered triple of ordered pairs. (Contributed by NM, 14-Sep-2011.) |
Theorem | cnvcnvsn 4920 | Double converse of a singleton of an ordered pair. (Unlike cnvsn 4926, this does not need any sethood assumptions on and .) (Contributed by Mario Carneiro, 26-Apr-2015.) |
Theorem | dmsnsnsng 4921 | The domain of the singleton of the singleton of a singleton. (Contributed by Jim Kingdon, 16-Dec-2018.) |
Theorem | rnsnopg 4922 | 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 4923 | The range of a pair of ordered pairs is the pair of second members. (Contributed by Thierry Arnoux, 3-Jan-2017.) |
Theorem | rnsnop 4924 | 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 4925 | Extract the first member of an ordered pair. (See op2nda 4928 to extract the second member and op1stb 4313 for an alternate version.) (Contributed by Raph Levien, 4-Dec-2003.) |
Theorem | cnvsn 4926 | Converse of a singleton of an ordered pair. (Contributed by NM, 11-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Theorem | op2ndb 4927 | Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4313 to extract the first member and op2nda 4928 for an alternate version.) (Contributed by NM, 25-Nov-2003.) |
Theorem | op2nda 4928 | Extract the second member of an ordered pair. (See op1sta 4925 to extract the first member and op2ndb 4927 for an alternate version.) (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | cnvsng 4929 | Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.) |
Theorem | opswapg 4930 | Swap the members of an ordered pair. (Contributed by Jim Kingdon, 16-Dec-2018.) |
Theorem | elxp4 4931 | Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 4932. (Contributed by NM, 17-Feb-2004.) |
Theorem | elxp5 4932 | Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 4931 when the double intersection does not create class existence problems (caused by int0 3708). (Contributed by NM, 1-Aug-2004.) |
Theorem | cnvresima 4933 | An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.) |
Theorem | resdm2 4934 | A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.) |
Theorem | resdmres 4935 | Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.) |
Theorem | imadmres 4936 | The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.) |
Theorem | mptpreima 4937* | The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
Theorem | mptiniseg 4938* | Converse singleton image of a function defined by maps-to. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
Theorem | dmmpt 4939 | The domain of the mapping operation in general. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 22-Mar-2015.) |
Theorem | dmmptss 4940* | The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.) |
Theorem | dmmptg 4941* | 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 4942 | A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.) |
Theorem | dfco2 4943* | Alternate definition of a class composition, using only one bound variable. (Contributed by NM, 19-Dec-2008.) |
Theorem | dfco2a 4944* | Generalization of dfco2 4943, where can have any value between and . (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | coundi 4945 | Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | coundir 4946 | Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | cores 4947 | Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | resco 4948 | Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.) |
Theorem | imaco 4949 | Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.) |
Theorem | rnco 4950 | The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.) |
Theorem | rnco2 4951 | The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.) |
Theorem | dmco 4952 | The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.) |
Theorem | coiun 4953* | Composition with an indexed union. (Contributed by NM, 21-Dec-2008.) |
Theorem | cocnvcnv1 4954 | A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.) |
Theorem | cocnvcnv2 4955 | A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.) |
Theorem | cores2 4956 | Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.) |
Theorem | co02 4957 | Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.) |
Theorem | co01 4958 | Composition with the empty set. (Contributed by NM, 24-Apr-2004.) |
Theorem | coi1 4959 | Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.) |
Theorem | coi2 4960 | Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.) |
Theorem | coires1 4961 | Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.) |
Theorem | coass 4962 | 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 4963 | A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.) |
Theorem | relssdmrn 4964 | 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 4965 | The converse is a subset of the cartesian product of range and domain. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Theorem | cossxp 4966 | Composition as a subset of the cross product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.) |
Theorem | cossxp2 4967 | The composition of two relations is a relation, with bounds on its domain and codomain. (Contributed by BJ, 10-Jul-2022.) |
Theorem | cocnvres 4968 | Restricting a relation and a converse relation when they are composed together (Contributed by BJ, 10-Jul-2022.) |
Theorem | cocnvss 4969 | Upper bound for the composed of a relation and an inverse relation. (Contributed by BJ, 10-Jul-2022.) |
Theorem | relrelss 4970 | Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.) |
Theorem | unielrel 4971 | The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.) |
Theorem | relfld 4972 | The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.) |
Theorem | relresfld 4973 | Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.) |
Theorem | relcoi2 4974 | Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.) |
Theorem | relcoi1 4975 | Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.) |
Theorem | unidmrn 4976 | The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.) |
Theorem | relcnvfld 4977 | if is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.) |
Theorem | dfdm2 4978 | Alternate definition of domain df-dm 4462 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.) |
Theorem | unixpm 4979* | The double class union of an inhabited cross product is the union of its members. (Contributed by Jim Kingdon, 18-Dec-2018.) |
Theorem | unixp0im 4980 | The union of an empty cross product is empty. (Contributed by Jim Kingdon, 18-Dec-2018.) |
Theorem | cnvexg 4981 | The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.) |
Theorem | cnvex 4982 | The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 19-Dec-2003.) |
Theorem | relcnvexb 4983 | A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.) |
Theorem | ressn 4984 | Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) |
Theorem | cnviinm 4985* | The converse of an intersection is the intersection of the converse. (Contributed by Jim Kingdon, 18-Dec-2018.) |
Theorem | cnvpom 4986* | The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.) |
Theorem | cnvsom 4987* | The converse of a strict order relation is a strict order relation. (Contributed by Jim Kingdon, 19-Dec-2018.) |
Theorem | coexg 4988 | The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.) |
Theorem | coex 4989 | The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.) |
Theorem | xpcom 4990* | Composition of two cross products. (Contributed by Jim Kingdon, 20-Dec-2018.) |
Syntax | cio 4991 | Extend class notation with Russell's definition description binder (inverted iota). |
Theorem | iotajust 4992* | Soundness justification theorem for df-iota 4993. (Contributed by Andrew Salmon, 29-Jun-2011.) |
Definition | df-iota 4993* |
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 5004);
otherwise, it evaluates to the empty set (see iotanul 5008). 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 5016 (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 4994* | Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.) |
Theorem | nfiota1 4995 | Bound-variable hypothesis builder for the class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Theorem | nfiotadxy 4996* | Deduction version of nfiotaxy 4997. (Contributed by Jim Kingdon, 21-Dec-2018.) |
Theorem | nfiotaxy 4997* | Bound-variable hypothesis builder for the class. (Contributed by NM, 23-Aug-2011.) |
Theorem | cbviota 4998 | Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.) |
Theorem | cbviotav 4999* | Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.) |
Theorem | sb8iota 5000 | Variable substitution in description binder. Compare sb8eu 1962. (Contributed by NM, 18-Mar-2013.) |
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