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

Theoremiss 4801 A subclass of the identity function is the identity function restricted to its domain. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremresopab2 4802* Restriction of a class abstraction of ordered pairs. (Contributed by NM, 24-Aug-2007.)

Theoremresmpt 4803* Restriction of the mapping operation. (Contributed by Mario Carneiro, 15-Jul-2013.)

Theoremresmpt3 4804* Unconditional restriction of the mapping operation. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Proof shortened by Mario Carneiro, 22-Mar-2015.)

Theoremresmptf 4805 Restriction of the mapping operation. (Contributed by Thierry Arnoux, 28-Mar-2017.)

Theoremresmptd 4806* Restriction of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremdfres2 4807* Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.)

Theoremopabresid 4808* The restricted identity expressed with the class builder. (Contributed by FL, 25-Apr-2012.)

Theoremmptresid 4809* The restricted identity expressed with the maps-to notation. (Contributed by FL, 25-Apr-2012.)

Theoremdmresi 4810 The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.)

Theoremresid 4811 Any relation restricted to the universe is itself. (Contributed by NM, 16-Mar-2004.)

Theoremimaeq1 4812 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Theoremimaeq2 4813 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Theoremimaeq1i 4814 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)

Theoremimaeq2i 4815 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)

Theoremimaeq1d 4816 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)

Theoremimaeq2d 4817 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)

Theoremimaeq12d 4818 Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.)

Theoremdfima2 4819* Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 19-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremdfima3 4820* Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremelimag 4821* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 20-Jan-2007.)

Theoremelima 4822* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 19-Apr-2004.)

Theoremelima2 4823* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 11-Aug-2004.)

Theoremelima3 4824* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 14-Aug-1994.)

Theoremnfima 4825 Bound-variable hypothesis builder for image. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremnfimad 4826 Deduction version of bound-variable hypothesis builder nfima 4825. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremimadmrn 4827 The image of the domain of a class is the range of the class. (Contributed by NM, 14-Aug-1994.)

Theoremimassrn 4828 The image of a class is a subset of its range. Theorem 3.16(xi) of [Monk1] p. 39. (Contributed by NM, 31-Mar-1995.)

Theoremimaexg 4829 The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by NM, 24-Jul-1995.)

Theoremimaex 4830 The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by JJ, 24-Sep-2021.)

Theoremimai 4831 Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.)

Theoremrnresi 4832 The range of the restricted identity function. (Contributed by NM, 27-Aug-2004.)

Theoremresiima 4833 The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)

Theoremima0 4834 Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.)

Theorem0ima 4835 Image under the empty relation. (Contributed by FL, 11-Jan-2007.)

Theoremcsbima12g 4836 Move class substitution in and out of the image of a function. (Contributed by FL, 15-Dec-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)

Theoremimadisj 4837 A class whose image under another is empty is disjoint with the other's domain. (Contributed by FL, 24-Jan-2007.)

Theoremcnvimass 4838 A preimage under any class is included in the domain of the class. (Contributed by FL, 29-Jan-2007.)

Theoremcnvimarndm 4839 The preimage of the range of a class is the domain of the class. (Contributed by Jeff Hankins, 15-Jul-2009.)

Theoremimasng 4840* The image of a singleton. (Contributed by NM, 8-May-2005.)

Theoremelreimasng 4841 Elementhood in the image of a singleton. (Contributed by Jim Kingdon, 10-Dec-2018.)

Theoremelimasn 4842 Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremelimasng 4843 Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.)

Theoremargs 4844* Two ways to express the class of unique-valued arguments of , which is the same as the domain of whenever is a function. The left-hand side of the equality is from Definition 10.2 of [Quine] p. 65. Quine uses the notation "arg " for this class (for which we have no separate notation). (Contributed by NM, 8-May-2005.)

Theoremeliniseg 4845 Membership in an initial segment. The idiom , meaning , is used to specify an initial segment in (for example) Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 28-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremepini 4846 Any set is equal to its preimage under the converse epsilon relation. (Contributed by Mario Carneiro, 9-Mar-2013.)

Theoreminiseg 4847* An idiom that signifies an initial segment of an ordering, used, for example, in Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 28-Apr-2004.)

Theoremdfse2 4848* Alternate definition of set-like relation. (Contributed by Mario Carneiro, 23-Jun-2015.)
Se

Theoremexse2 4849 Any set relation is set-like. (Contributed by Mario Carneiro, 22-Jun-2015.)
Se

Theoremimass1 4850 Subset theorem for image. (Contributed by NM, 16-Mar-2004.)

Theoremimass2 4851 Subset theorem for image. Exercise 22(a) of [Enderton] p. 53. (Contributed by NM, 22-Mar-1998.)

Theoremndmima 4852 The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998.)

Theoremrelcnv 4853 A converse is a relation. Theorem 12 of [Suppes] p. 62. (Contributed by NM, 29-Oct-1996.)

Theoremrelbrcnvg 4854 When is a relation, the sethood assumptions on brcnv 4660 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)

Theoremrelbrcnv 4855 When is a relation, the sethood assumptions on brcnv 4660 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)

Theoremcotr 4856* Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremissref 4857* Two ways to state a relation is reflexive. Adapted from Tarski. (Contributed by FL, 15-Jan-2012.) (Revised by NM, 30-Mar-2016.)

Theoremcnvsym 4858* Two ways of saying a relation is symmetric. Similar to definition of symmetry in [Schechter] p. 51. (Contributed by NM, 28-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremintasym 4859* Two ways of saying a relation is antisymmetric. Definition of antisymmetry in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremasymref 4860* Two ways of saying a relation is antisymmetric and reflexive. is the field of a relation by relfld 5003. (Contributed by NM, 6-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremintirr 4861* Two ways of saying a relation is irreflexive. Definition of irreflexivity in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theorembrcodir 4862* Two ways of saying that two elements have an upper bound. (Contributed by Mario Carneiro, 3-Nov-2015.)

Theoremcodir 4863* Two ways of saying a relation is directed. (Contributed by Mario Carneiro, 22-Nov-2013.)

Theoremqfto 4864* A quantifier-free way of expressing the total order predicate. (Contributed by Mario Carneiro, 22-Nov-2013.)

Theoremxpidtr 4865 A square cross product is a transitive relation. (Contributed by FL, 31-Jul-2009.)

Theoremtrin2 4866 The intersection of two transitive classes is transitive. (Contributed by FL, 31-Jul-2009.)

Theorempoirr2 4867 A partial order relation is irreflexive. (Contributed by Mario Carneiro, 2-Nov-2015.)

Theoremtrinxp 4868 The relation induced by a transitive relation on a part of its field is transitive. (Taking the intersection of a relation with a square cross product is a way to restrict it to a subset of its field.) (Contributed by FL, 31-Jul-2009.)

Theoremsoirri 4869 A strict order relation is irreflexive. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)

Theoremsotri 4870 A strict order relation is a transitive relation. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)

Theoremson2lpi 4871 A strict order relation has no 2-cycle loops. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)

Theoremsotri2 4872 A transitivity relation. (Read B < A and B < C implies A < C .) (Contributed by Mario Carneiro, 10-May-2013.)

Theoremsotri3 4873 A transitivity relation. (Read A < B and C < B implies A < C .) (Contributed by Mario Carneiro, 10-May-2013.)

Theorempoleloe 4874 Express "less than or equals" for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theorempoltletr 4875 Transitive law for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremcnvopab 4876* The converse of a class abstraction of ordered pairs. (Contributed by NM, 11-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremmptcnv 4877* The converse of a mapping function. (Contributed by Thierry Arnoux, 16-Jan-2017.)

Theoremcnv0 4878 The converse of the empty set. (Contributed by NM, 6-Apr-1998.)

Theoremcnvi 4879 The converse of the identity relation. Theorem 3.7(ii) of [Monk1] p. 36. (Contributed by NM, 26-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcnvun 4880 The converse of a union is the union of converses. Theorem 16 of [Suppes] p. 62. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcnvdif 4881 Distributive law for converse over set difference. (Contributed by Mario Carneiro, 26-Jun-2014.)

Theoremcnvin 4882 Distributive law for converse over intersection. Theorem 15 of [Suppes] p. 62. (Contributed by NM, 25-Mar-1998.) (Revised by Mario Carneiro, 26-Jun-2014.)

Theoremrnun 4883 Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.)

Theoremrnin 4884 The range of an intersection belongs the intersection of ranges. Theorem 9 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)

Theoremrniun 4885 The range of an indexed union. (Contributed by Mario Carneiro, 29-May-2015.)

Theoremrnuni 4886* The range of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by NM, 17-Mar-2004.) (Revised by Mario Carneiro, 29-May-2015.)

Theoremimaundi 4887 Distributive law for image over union. Theorem 35 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)

Theoremimaundir 4888 The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)

Theoremdminss 4889 An upper bound for intersection with a domain. Theorem 40 of [Suppes] p. 66, who calls it "somewhat surprising." (Contributed by NM, 11-Aug-2004.)

Theoremimainss 4890 An upper bound for intersection with an image. Theorem 41 of [Suppes] p. 66. (Contributed by NM, 11-Aug-2004.)

Theoreminimass 4891 The image of an intersection (Contributed by Thierry Arnoux, 16-Dec-2017.)

Theoreminimasn 4892 The intersection of the image of singleton (Contributed by Thierry Arnoux, 16-Dec-2017.)

Theoremcnvxp 4893 The converse of a cross product. Exercise 11 of [Suppes] p. 67. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremxp0 4894 The cross product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.)

Theoremxpmlem 4895* The cross product of inhabited classes is inhabited. (Contributed by Jim Kingdon, 11-Dec-2018.)

Theoremxpm 4896* The cross product of inhabited classes is inhabited. (Contributed by Jim Kingdon, 13-Dec-2018.)

Theoremxpeq0r 4897 A cross product is empty if at least one member is empty. (Contributed by Jim Kingdon, 12-Dec-2018.)

Theoremsqxpeq0 4898 A Cartesian square is empty iff its member is empty. (Contributed by Jim Kingdon, 21-Apr-2023.)

Theoremxpdisj1 4899 Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)

Theoremxpdisj2 4900 Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)

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