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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | rnsnop 4801 | 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 4802 | Extract the first member of an ordered pair. (See op2nda 4805 to extract the second member and op1stb 4209 for an alternate version.) (Contributed by Raph Levien, 4-Dec-2003.) |
Theorem | cnvsn 4803 | Converse of a singleton of an ordered pair. (Contributed by NM, 11-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Theorem | op2ndb 4804 | Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4209 to extract the first member and op2nda 4805 for an alternate version.) (Contributed by NM, 25-Nov-2003.) |
Theorem | op2nda 4805 | Extract the second member of an ordered pair. (See op1sta 4802 to extract the first member and op2ndb 4804 for an alternate version.) (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | cnvsng 4806 | Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.) |
Theorem | opswapg 4807 | Swap the members of an ordered pair. (Contributed by Jim Kingdon, 16-Dec-2018.) |
Theorem | elxp4 4808 | Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 4809. (Contributed by NM, 17-Feb-2004.) |
Theorem | elxp5 4809 | Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 4808 when the double intersection does not create class existence problems (caused by int0 3629). (Contributed by NM, 1-Aug-2004.) |
Theorem | cnvresima 4810 | An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.) |
Theorem | resdm2 4811 | A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.) |
Theorem | resdmres 4812 | Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.) |
Theorem | imadmres 4813 | The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.) |
Theorem | mptpreima 4814* | The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
Theorem | mptiniseg 4815* | Converse singleton image of a function defined by maps-to. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
Theorem | dmmpt 4816 | The domain of the mapping operation in general. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 22-Mar-2015.) |
Theorem | dmmptss 4817* | The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.) |
Theorem | dmmptg 4818* | 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 4819 | A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.) |
Theorem | dfco2 4820* | Alternate definition of a class composition, using only one bound variable. (Contributed by NM, 19-Dec-2008.) |
Theorem | dfco2a 4821* | Generalization of dfco2 4820, where can have any value between and . (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | coundi 4822 | Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | coundir 4823 | Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | cores 4824 | Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | resco 4825 | Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.) |
Theorem | imaco 4826 | Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.) |
Theorem | rnco 4827 | The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.) |
Theorem | rnco2 4828 | The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.) |
Theorem | dmco 4829 | The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.) |
Theorem | coiun 4830* | Composition with an indexed union. (Contributed by NM, 21-Dec-2008.) |
Theorem | cocnvcnv1 4831 | A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.) |
Theorem | cocnvcnv2 4832 | A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.) |
Theorem | cores2 4833 | Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.) |
Theorem | co02 4834 | Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.) |
Theorem | co01 4835 | Composition with the empty set. (Contributed by NM, 24-Apr-2004.) |
Theorem | coi1 4836 | Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.) |
Theorem | coi2 4837 | Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.) |
Theorem | coires1 4838 | Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.) |
Theorem | coass 4839 | 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 4840 | A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.) |
Theorem | relssdmrn 4841 | 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 4842 | The converse is a subset of the cartesian product of range and domain. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Theorem | cossxp 4843 | Composition as a subset of the cross product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.) |
Theorem | relrelss 4844 | Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.) |
Theorem | unielrel 4845 | The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.) |
Theorem | relfld 4846 | The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.) |
Theorem | relresfld 4847 | Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.) |
Theorem | relcoi2 4848 | Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.) |
Theorem | relcoi1 4849 | Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.) |
Theorem | unidmrn 4850 | The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.) |
Theorem | relcnvfld 4851 | if is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.) |
Theorem | dfdm2 4852 | Alternate definition of domain df-dm 4355 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.) |
Theorem | unixpm 4853* | The double class union of an inhabited cross product is the union of its members. (Contributed by Jim Kingdon, 18-Dec-2018.) |
Theorem | unixp0im 4854 | The union of an empty cross product is empty. (Contributed by Jim Kingdon, 18-Dec-2018.) |
Theorem | cnvexg 4855 | The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.) |
Theorem | cnvex 4856 | The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 19-Dec-2003.) |
Theorem | relcnvexb 4857 | A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.) |
Theorem | ressn 4858 | Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) |
Theorem | cnviinm 4859* | The converse of an intersection is the intersection of the converse. (Contributed by Jim Kingdon, 18-Dec-2018.) |
Theorem | cnvpom 4860* | The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.) |
Theorem | cnvsom 4861* | The converse of a strict order relation is a strict order relation. (Contributed by Jim Kingdon, 19-Dec-2018.) |
Theorem | coexg 4862 | The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.) |
Theorem | coex 4863 | The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.) |
Theorem | xpcom 4864* | Composition of two cross products. (Contributed by Jim Kingdon, 20-Dec-2018.) |
Syntax | cio 4865 | Extend class notation with Russell's definition description binder (inverted iota). |
Theorem | iotajust 4866* | Soundness justification theorem for df-iota 4867. (Contributed by Andrew Salmon, 29-Jun-2011.) |
Definition | df-iota 4867* |
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 4878);
otherwise, it evaluates to the empty set (see iotanul 4882). 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 4890 (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 4868* | Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.) |
Theorem | nfiota1 4869 | Bound-variable hypothesis builder for the class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Theorem | nfiotadxy 4870* | Deduction version of nfiotaxy 4871. (Contributed by Jim Kingdon, 21-Dec-2018.) |
Theorem | nfiotaxy 4871* | Bound-variable hypothesis builder for the class. (Contributed by NM, 23-Aug-2011.) |
Theorem | cbviota 4872 | Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.) |
Theorem | cbviotav 4873* | Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.) |
Theorem | sb8iota 4874 | Variable substitution in description binder. Compare sb8eu 1913. (Contributed by NM, 18-Mar-2013.) |
Theorem | iotaeq 4875 | Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.) |
Theorem | iotabi 4876 | Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.) |
Theorem | uniabio 4877* | 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 4878* | Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.) |
Theorem | iotauni 4879 | Equivalence between two different forms of . (Contributed by Andrew Salmon, 12-Jul-2011.) |
Theorem | iotaint 4880 | Equivalence between two different forms of . (Contributed by Mario Carneiro, 24-Dec-2016.) |
Theorem | iota1 4881 | Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) |
Theorem | iotanul 4882 | 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 4883 | 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 4884* | 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 4885 | Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.) |
Theorem | iota4an 4886 | Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.) |
Theorem | iota5 4887* | A method for computing iota. (Contributed by NM, 17-Sep-2013.) |
Theorem | iotabidv 4888* | Formula-building deduction rule for iota. (Contributed by NM, 20-Aug-2011.) |
Theorem | iotabii 4889 | Formula-building deduction rule for iota. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Theorem | iotacl 4890 |
Membership law for descriptions.
This can useful for expanding an unbounded iota-based definition (see df-iota 4867). (Contributed by Andrew Salmon, 1-Aug-2011.) |
Theorem | iota2df 4891 | A condition that allows us to represent "the unique element such that " with a class expression . (Contributed by NM, 30-Dec-2014.) |
Theorem | iota2d 4892* | A condition that allows us to represent "the unique element such that " with a class expression . (Contributed by NM, 30-Dec-2014.) |
Theorem | iota2 4893* | The unique element such that . (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) |
Theorem | sniota 4894 | A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | csbiotag 4895* | Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.) |
Syntax | wfun 4896 | Extend the definition of a wff to include the function predicate. (Read: is a function.) |
Syntax | wfn 4897 | Extend the definition of a wff to include the function predicate with a domain. (Read: is a function on .) |
Syntax | wf 4898 | Extend the definition of a wff to include the function predicate with domain and codomain. (Read: maps into .) |
Syntax | wf1 4899 | 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 4900 | 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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