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Theorem List for Intuitionistic Logic Explorer - 4801-4900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrnsnop 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.)

Theoremop1sta 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.)

Theoremcnvsn 4803 Converse of a singleton of an ordered pair. (Contributed by NM, 11-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremop2ndb 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.)

Theoremop2nda 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.)

Theoremcnvsng 4806 Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.)

Theoremopswapg 4807 Swap the members of an ordered pair. (Contributed by Jim Kingdon, 16-Dec-2018.)

Theoremelxp4 4808 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 4809. (Contributed by NM, 17-Feb-2004.)

Theoremelxp5 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.)

Theoremcnvresima 4810 An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.)

Theoremresdm2 4811 A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.)

Theoremresdmres 4812 Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.)

Theoremimadmres 4813 The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.)

Theoremmptpreima 4814* The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Theoremmptiniseg 4815* Converse singleton image of a function defined by maps-to. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Theoremdmmpt 4816 The domain of the mapping operation in general. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 22-Mar-2015.)

Theoremdmmptss 4817* The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.)

Theoremdmmptg 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.)

Theoremrelco 4819 A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.)

Theoremdfco2 4820* Alternate definition of a class composition, using only one bound variable. (Contributed by NM, 19-Dec-2008.)

Theoremdfco2a 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.)

Theoremcoundi 4822 Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcoundir 4823 Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcores 4824 Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremresco 4825 Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)

Theoremimaco 4826 Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.)

Theoremrnco 4827 The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.)

Theoremrnco2 4828 The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.)

Theoremdmco 4829 The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.)

Theoremcoiun 4830* Composition with an indexed union. (Contributed by NM, 21-Dec-2008.)

Theoremcocnvcnv1 4831 A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.)

Theoremcocnvcnv2 4832 A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.)

Theoremcores2 4833 Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.)

Theoremco02 4834 Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.)

Theoremco01 4835 Composition with the empty set. (Contributed by NM, 24-Apr-2004.)

Theoremcoi1 4836 Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)

Theoremcoi2 4837 Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)

Theoremcoires1 4838 Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.)

Theoremcoass 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.)

Theoremrelcnvtr 4840 A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.)

Theoremrelssdmrn 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.)

Theoremcnvssrndm 4842 The converse is a subset of the cartesian product of range and domain. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremcossxp 4843 Composition as a subset of the cross product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.)

Theoremrelrelss 4844 Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.)

Theoremunielrel 4845 The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.)

Theoremrelfld 4846 The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.)

Theoremrelresfld 4847 Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.)

Theoremrelcoi2 4848 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.)

Theoremrelcoi1 4849 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.)

Theoremunidmrn 4850 The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.)

Theoremrelcnvfld 4851 if is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.)

Theoremdfdm2 4852 Alternate definition of domain df-dm 4355 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.)

Theoremunixpm 4853* The double class union of an inhabited cross product is the union of its members. (Contributed by Jim Kingdon, 18-Dec-2018.)

Theoremunixp0im 4854 The union of an empty cross product is empty. (Contributed by Jim Kingdon, 18-Dec-2018.)

Theoremcnvexg 4855 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.)

Theoremcnvex 4856 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 19-Dec-2003.)

Theoremrelcnvexb 4857 A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.)

Theoremressn 4858 Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)

Theoremcnviinm 4859* The converse of an intersection is the intersection of the converse. (Contributed by Jim Kingdon, 18-Dec-2018.)

Theoremcnvpom 4860* The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.)

Theoremcnvsom 4861* The converse of a strict order relation is a strict order relation. (Contributed by Jim Kingdon, 19-Dec-2018.)

Theoremcoexg 4862 The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.)

Theoremcoex 4863 The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.)

Theoremxpcom 4864* Composition of two cross products. (Contributed by Jim Kingdon, 20-Dec-2018.)

2.6.7  Definite description binder (inverted iota)

Syntaxcio 4865 Extend class notation with Russell's definition description binder (inverted iota).

Theoremiotajust 4866* Soundness justification theorem for df-iota 4867. (Contributed by Andrew Salmon, 29-Jun-2011.)

Definitiondf-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.)

Theoremdfiota2 4868* Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.)

Theoremnfiota1 4869 Bound-variable hypothesis builder for the class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremnfiotadxy 4870* Deduction version of nfiotaxy 4871. (Contributed by Jim Kingdon, 21-Dec-2018.)

Theoremnfiotaxy 4871* Bound-variable hypothesis builder for the class. (Contributed by NM, 23-Aug-2011.)

Theoremcbviota 4872 Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)

Theoremcbviotav 4873* Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)

Theoremsb8iota 4874 Variable substitution in description binder. Compare sb8eu 1913. (Contributed by NM, 18-Mar-2013.)

Theoremiotaeq 4875 Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)

Theoremiotabi 4876 Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)

Theoremuniabio 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.)

Theoremiotaval 4878* Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)

Theoremiotauni 4879 Equivalence between two different forms of . (Contributed by Andrew Salmon, 12-Jul-2011.)

Theoremiotaint 4880 Equivalence between two different forms of . (Contributed by Mario Carneiro, 24-Dec-2016.)

Theoremiota1 4881 Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)

Theoremiotanul 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.)

Theoremeuiotaex 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.)

Theoremiotass 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.)

Theoremiota4 4885 Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)

Theoremiota4an 4886 Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)

Theoremiota5 4887* A method for computing iota. (Contributed by NM, 17-Sep-2013.)

Theoremiotabidv 4888* Formula-building deduction rule for iota. (Contributed by NM, 20-Aug-2011.)

Theoremiotabii 4889 Formula-building deduction rule for iota. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremiotacl 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.)

Theoremiota2df 4891 A condition that allows us to represent "the unique element such that " with a class expression . (Contributed by NM, 30-Dec-2014.)

Theoremiota2d 4892* A condition that allows us to represent "the unique element such that " with a class expression . (Contributed by NM, 30-Dec-2014.)

Theoremiota2 4893* The unique element such that . (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)

Theoremsniota 4894 A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremcsbiotag 4895* Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.)

2.6.8  Functions

Syntaxwfun 4896 Extend the definition of a wff to include the function predicate. (Read: is a function.)

Syntaxwfn 4897 Extend the definition of a wff to include the function predicate with a domain. (Read: is a function on .)

Syntaxwf 4898 Extend the definition of a wff to include the function predicate with domain and codomain. (Read: maps into .)

Syntaxwf1 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.

Syntaxwfo 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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