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

Theoremdfiun3 4801 Alternate definition of indexed union when is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremdfiin3 4802 Alternate definition of indexed intersection when is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremriinint 4803* Express a relative indexed intersection as an intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremrelrn0 4804 A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004.)

Theoremdmrnssfld 4805 The domain and range of a class are included in its double union. (Contributed by NM, 13-May-2008.)

Theoremdmexg 4806 The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Apr-1995.)

Theoremrnexg 4807 The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26. Similar to Lemma 3D of [Enderton] p. 41. (Contributed by NM, 31-Mar-1995.)

Theoremdmex 4808 The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Jul-2008.)

Theoremrnex 4809 The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26. Similar to Lemma 3D of [Enderton] p. 41. (Contributed by NM, 7-Jul-2008.)

Theoremiprc 4810 The identity function is a proper class. This means, for example, that we cannot use it as a member of the class of continuous functions unless it is restricted to a set. (Contributed by NM, 1-Jan-2007.)

Theoremdmcoss 4811 Domain of a composition. Theorem 21 of [Suppes] p. 63. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremrncoss 4812 Range of a composition. (Contributed by NM, 19-Mar-1998.)

Theoremdmcosseq 4813 Domain of a composition. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremdmcoeq 4814 Domain of a composition. (Contributed by NM, 19-Mar-1998.)

Theoremrncoeq 4815 Range of a composition. (Contributed by NM, 19-Mar-1998.)

Theoremreseq1 4816 Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994.)

Theoremreseq2 4817 Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.)

Theoremreseq1i 4818 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)

Theoremreseq2i 4819 Equality inference for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremreseq12i 4820 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)

Theoremreseq1d 4821 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)

Theoremreseq2d 4822 Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremreseq12d 4823 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)

Theoremnfres 4824 Bound-variable hypothesis builder for restriction. (Contributed by NM, 15-Sep-2003.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremcsbresg 4825 Distribute proper substitution through the restriction of a class. (Contributed by Alan Sare, 10-Nov-2012.)

Theoremres0 4826 A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.)

Theoremopelres 4827 Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.)

Theorembrres 4828 Binary relation on a restriction. (Contributed by NM, 12-Dec-2006.)

Theoremopelresg 4829 Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 14-Oct-2005.)

Theorembrresg 4830 Binary relation on a restriction. (Contributed by Mario Carneiro, 4-Nov-2015.)

Theoremopres 4831 Ordered pair membership in a restriction when the first member belongs to the restricting class. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremresieq 4832 A restricted identity relation is equivalent to equality in its domain. (Contributed by NM, 30-Apr-2004.)

Theoremopelresi 4833 belongs to a restriction of the identity class iff belongs to the restricting class. (Contributed by FL, 27-Oct-2008.) (Revised by NM, 30-Mar-2016.)

Theoremresres 4834 The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)

Theoremresundi 4835 Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)

Theoremresundir 4836 Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.)

Theoremresindi 4837 Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)

Theoremresindir 4838 Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.)

Theoreminres 4839 Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.)

Theoremresdifcom 4840 Commutative law for restriction and difference. (Contributed by AV, 7-Jun-2021.)

Theoremresiun1 4841* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)

Theoremresiun2 4842* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)

Theoremdmres 4843 The domain of a restriction. Exercise 14 of [TakeutiZaring] p. 25. (Contributed by NM, 1-Aug-1994.)

Theoremssdmres 4844 A domain restricted to a subclass equals the subclass. (Contributed by NM, 2-Mar-1997.)

Theoremdmresexg 4845 The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995.)

Theoremresss 4846 A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)

Theoremrescom 4847 Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)

Theoremssres 4848 Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.)

Theoremssres2 4849 Subclass theorem for restriction. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremrelres 4850 A restriction is a relation. Exercise 12 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremresabs1 4851 Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25. (Contributed by NM, 9-Aug-1994.)

Theoremresabs1d 4852 Absorption law for restriction, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremresabs2 4853 Absorption law for restriction. (Contributed by NM, 27-Mar-1998.)

Theoremresidm 4854 Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.)

Theoremresima 4855 A restriction to an image. (Contributed by NM, 29-Sep-2004.)

Theoremresima2 4856 Image under a restricted class. (Contributed by FL, 31-Aug-2009.)

Theoremxpssres 4857 Restriction of a constant function (or other cross product). (Contributed by Stefan O'Rear, 24-Jan-2015.)

Theoremelres 4858* Membership in a restriction. (Contributed by Scott Fenton, 17-Mar-2011.)

Theoremelsnres 4859* Memebership in restriction to a singleton. (Contributed by Scott Fenton, 17-Mar-2011.)

Theoremrelssres 4860 Simplification law for restriction. (Contributed by NM, 16-Aug-1994.)

Theoremresdm 4861 A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.)

Theoremresexg 4862 The restriction of a set is a set. (Contributed by NM, 28-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremresex 4863 The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.)

Theoremresindm 4864 When restricting a relation, intersecting with the domain of the relation has no effect. (Contributed by FL, 6-Oct-2008.)

Theoremresdmdfsn 4865 Restricting a relation to its domain without a set is the same as restricting the relation to the universe without this set. (Contributed by AV, 2-Dec-2018.)

Theoremresopab 4866* Restriction of a class abstraction of ordered pairs. (Contributed by NM, 5-Nov-2002.)

Theoremresiexg 4867 The existence of a restricted identity function, proved without using the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.)

Theoremiss 4868 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 4869* Restriction of a class abstraction of ordered pairs. (Contributed by NM, 24-Aug-2007.)

Theoremresmpt 4870* Restriction of the mapping operation. (Contributed by Mario Carneiro, 15-Jul-2013.)

Theoremresmpt3 4871* Unconditional restriction of the mapping operation. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Proof shortened by Mario Carneiro, 22-Mar-2015.)

Theoremresmptf 4872 Restriction of the mapping operation. (Contributed by Thierry Arnoux, 28-Mar-2017.)

Theoremresmptd 4873* Restriction of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremdfres2 4874* Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.)

Theoremopabresid 4875* The restricted identity expressed with the class builder. (Contributed by FL, 25-Apr-2012.)

Theoremmptresid 4876* The restricted identity expressed with the maps-to notation. (Contributed by FL, 25-Apr-2012.)

Theoremdmresi 4877 The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.)

Theoremresid 4878 Any relation restricted to the universe is itself. (Contributed by NM, 16-Mar-2004.)

Theoremimaeq1 4879 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Theoremimaeq2 4880 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Theoremimaeq1i 4881 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)

Theoremimaeq2i 4882 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)

Theoremimaeq1d 4883 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)

Theoremimaeq2d 4884 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)

Theoremimaeq12d 4885 Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.)

Theoremdfima2 4886* 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 4887* 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 4888* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 20-Jan-2007.)

Theoremelima 4889* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 19-Apr-2004.)

Theoremelima2 4890* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 11-Aug-2004.)

Theoremelima3 4891* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 14-Aug-1994.)

Theoremnfima 4892 Bound-variable hypothesis builder for image. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremnfimad 4893 Deduction version of bound-variable hypothesis builder nfima 4892. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremimadmrn 4894 The image of the domain of a class is the range of the class. (Contributed by NM, 14-Aug-1994.)

Theoremimassrn 4895 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 4896 The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by NM, 24-Jul-1995.)

Theoremimaex 4897 The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by JJ, 24-Sep-2021.)

Theoremimai 4898 Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.)

Theoremrnresi 4899 The range of the restricted identity function. (Contributed by NM, 27-Aug-2004.)

Theoremresiima 4900 The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)

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