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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | op2ndb 4901 | Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4290 to extract the first member and op2nda 4902 for an alternate version.) (Contributed by NM, 25-Nov-2003.) |
Theorem | op2nda 4902 | Extract the second member of an ordered pair. (See op1sta 4899 to extract the first member and op2ndb 4901 for an alternate version.) (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | cnvsng 4903 | Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.) |
Theorem | opswapg 4904 | Swap the members of an ordered pair. (Contributed by Jim Kingdon, 16-Dec-2018.) |
Theorem | elxp4 4905 | Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 4906. (Contributed by NM, 17-Feb-2004.) |
Theorem | elxp5 4906 | Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 4905 when the double intersection does not create class existence problems (caused by int0 3697). (Contributed by NM, 1-Aug-2004.) |
Theorem | cnvresima 4907 | An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.) |
Theorem | resdm2 4908 | A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.) |
Theorem | resdmres 4909 | Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.) |
Theorem | imadmres 4910 | The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.) |
Theorem | mptpreima 4911* | The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
Theorem | mptiniseg 4912* | Converse singleton image of a function defined by maps-to. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
Theorem | dmmpt 4913 | The domain of the mapping operation in general. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 22-Mar-2015.) |
Theorem | dmmptss 4914* | The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.) |
Theorem | dmmptg 4915* | 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 4916 | A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.) |
Theorem | dfco2 4917* | Alternate definition of a class composition, using only one bound variable. (Contributed by NM, 19-Dec-2008.) |
Theorem | dfco2a 4918* | Generalization of dfco2 4917, where can have any value between and . (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | coundi 4919 | Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | coundir 4920 | Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | cores 4921 | Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | resco 4922 | Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.) |
Theorem | imaco 4923 | Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.) |
Theorem | rnco 4924 | The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.) |
Theorem | rnco2 4925 | The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.) |
Theorem | dmco 4926 | The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.) |
Theorem | coiun 4927* | Composition with an indexed union. (Contributed by NM, 21-Dec-2008.) |
Theorem | cocnvcnv1 4928 | A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.) |
Theorem | cocnvcnv2 4929 | A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.) |
Theorem | cores2 4930 | Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.) |
Theorem | co02 4931 | Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.) |
Theorem | co01 4932 | Composition with the empty set. (Contributed by NM, 24-Apr-2004.) |
Theorem | coi1 4933 | Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.) |
Theorem | coi2 4934 | Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.) |
Theorem | coires1 4935 | Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.) |
Theorem | coass 4936 | 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 4937 | A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.) |
Theorem | relssdmrn 4938 | 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 4939 | The converse is a subset of the cartesian product of range and domain. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Theorem | cossxp 4940 | Composition as a subset of the cross product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.) |
Theorem | cossxp2 4941 | The composition of two relations is a relation, with bounds on its domain and codomain. (Contributed by BJ, 10-Jul-2022.) |
Theorem | cocnvres 4942 | Restricting a relation and a converse relation when they are composed together (Contributed by BJ, 10-Jul-2022.) |
Theorem | cocnvss 4943 | Upper bound for the composed of a relation and an inverse relation. (Contributed by BJ, 10-Jul-2022.) |
Theorem | relrelss 4944 | Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.) |
Theorem | unielrel 4945 | The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.) |
Theorem | relfld 4946 | The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.) |
Theorem | relresfld 4947 | Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.) |
Theorem | relcoi2 4948 | Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.) |
Theorem | relcoi1 4949 | Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.) |
Theorem | unidmrn 4950 | The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.) |
Theorem | relcnvfld 4951 | if is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.) |
Theorem | dfdm2 4952 | Alternate definition of domain df-dm 4438 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.) |
Theorem | unixpm 4953* | The double class union of an inhabited cross product is the union of its members. (Contributed by Jim Kingdon, 18-Dec-2018.) |
Theorem | unixp0im 4954 | The union of an empty cross product is empty. (Contributed by Jim Kingdon, 18-Dec-2018.) |
Theorem | cnvexg 4955 | The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.) |
Theorem | cnvex 4956 | The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 19-Dec-2003.) |
Theorem | relcnvexb 4957 | A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.) |
Theorem | ressn 4958 | Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) |
Theorem | cnviinm 4959* | The converse of an intersection is the intersection of the converse. (Contributed by Jim Kingdon, 18-Dec-2018.) |
Theorem | cnvpom 4960* | The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.) |
Theorem | cnvsom 4961* | The converse of a strict order relation is a strict order relation. (Contributed by Jim Kingdon, 19-Dec-2018.) |
Theorem | coexg 4962 | The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.) |
Theorem | coex 4963 | The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.) |
Theorem | xpcom 4964* | Composition of two cross products. (Contributed by Jim Kingdon, 20-Dec-2018.) |
Syntax | cio 4965 | Extend class notation with Russell's definition description binder (inverted iota). |
Theorem | iotajust 4966* | Soundness justification theorem for df-iota 4967. (Contributed by Andrew Salmon, 29-Jun-2011.) |
Definition | df-iota 4967* |
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 4978);
otherwise, it evaluates to the empty set (see iotanul 4982). 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 4990 (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 4968* | Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.) |
Theorem | nfiota1 4969 | Bound-variable hypothesis builder for the class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Theorem | nfiotadxy 4970* | Deduction version of nfiotaxy 4971. (Contributed by Jim Kingdon, 21-Dec-2018.) |
Theorem | nfiotaxy 4971* | Bound-variable hypothesis builder for the class. (Contributed by NM, 23-Aug-2011.) |
Theorem | cbviota 4972 | Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.) |
Theorem | cbviotav 4973* | Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.) |
Theorem | sb8iota 4974 | Variable substitution in description binder. Compare sb8eu 1961. (Contributed by NM, 18-Mar-2013.) |
Theorem | iotaeq 4975 | Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.) |
Theorem | iotabi 4976 | Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.) |
Theorem | uniabio 4977* | 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 4978* | Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.) |
Theorem | iotauni 4979 | Equivalence between two different forms of . (Contributed by Andrew Salmon, 12-Jul-2011.) |
Theorem | iotaint 4980 | Equivalence between two different forms of . (Contributed by Mario Carneiro, 24-Dec-2016.) |
Theorem | iota1 4981 | Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) |
Theorem | iotanul 4982 | Theorem 8.22 in [Quine] p. 57. This theorem is the result if there isn't exactly one that satisfies . (Contributed by Andrew Salmon, 11-Jul-2011.) |
Theorem | euiotaex 4983 | Theorem 8.23 in [Quine] p. 58, with existential uniqueness condition added. This theorem proves the existence of the class under our definition. (Contributed by Jim Kingdon, 21-Dec-2018.) |
Theorem | iotass 4984* | Value of iota based on a proposition which holds only for values which are subsets of a given class. (Contributed by Mario Carneiro and Jim Kingdon, 21-Dec-2018.) |
Theorem | iota4 4985 | Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.) |
Theorem | iota4an 4986 | Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.) |
Theorem | iota5 4987* | A method for computing iota. (Contributed by NM, 17-Sep-2013.) |
Theorem | iotabidv 4988* | Formula-building deduction for iota. (Contributed by NM, 20-Aug-2011.) |
Theorem | iotabii 4989 | Formula-building deduction for iota. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Theorem | iotacl 4990 |
Membership law for descriptions.
This can useful for expanding an unbounded iota-based definition (see df-iota 4967). (Contributed by Andrew Salmon, 1-Aug-2011.) |
Theorem | iota2df 4991 | A condition that allows us to represent "the unique element such that " with a class expression . (Contributed by NM, 30-Dec-2014.) |
Theorem | iota2d 4992* | A condition that allows us to represent "the unique element such that " with a class expression . (Contributed by NM, 30-Dec-2014.) |
Theorem | iota2 4993* | The unique element such that . (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) |
Theorem | sniota 4994 | A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | csbiotag 4995* | Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.) |
Syntax | wfun 4996 | Extend the definition of a wff to include the function predicate. (Read: is a function.) |
Syntax | wfn 4997 | Extend the definition of a wff to include the function predicate with a domain. (Read: is a function on .) |
Syntax | wf 4998 | Extend the definition of a wff to include the function predicate with domain and codomain. (Read: maps into .) |
Syntax | wf1 4999 | Extend the definition of a wff to include one-to-one functions. (Read: maps one-to-one into .) The notation ("1-1" above the arrow) is from Definition 6.15(5) of [TakeutiZaring] p. 27. |
Syntax | wfo 5000 | Extend the definition of a wff to include onto functions. (Read: maps onto .) The notation ("onto" below the arrow) is from Definition 6.15(4) of [TakeutiZaring] p. 27. |
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