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Theorem List for Intuitionistic Logic Explorer - 4701-4800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremreseq2d 4701 Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremreseq12d 4702 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)

Theoremnfres 4703 Bound-variable hypothesis builder for restriction. (Contributed by NM, 15-Sep-2003.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremcsbresg 4704 Distribute proper substitution through the restriction of a class. (Contributed by Alan Sare, 10-Nov-2012.)

Theoremres0 4705 A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.)

Theoremopelres 4706 Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.)

Theorembrres 4707 Binary relation on a restriction. (Contributed by NM, 12-Dec-2006.)

Theoremopelresg 4708 Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 14-Oct-2005.)

Theorembrresg 4709 Binary relation on a restriction. (Contributed by Mario Carneiro, 4-Nov-2015.)

Theoremopres 4710 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 4711 A restricted identity relation is equivalent to equality in its domain. (Contributed by NM, 30-Apr-2004.)

Theoremopelresi 4712 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 4713 The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)

Theoremresundi 4714 Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)

Theoremresundir 4715 Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.)

Theoremresindi 4716 Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)

Theoremresindir 4717 Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.)

Theoreminres 4718 Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.)

Theoremresiun1 4719* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)

Theoremresiun2 4720* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)

Theoremdmres 4721 The domain of a restriction. Exercise 14 of [TakeutiZaring] p. 25. (Contributed by NM, 1-Aug-1994.)

Theoremssdmres 4722 A domain restricted to a subclass equals the subclass. (Contributed by NM, 2-Mar-1997.)

Theoremdmresexg 4723 The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995.)

Theoremresss 4724 A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)

Theoremrescom 4725 Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)

Theoremssres 4726 Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.)

Theoremssres2 4727 Subclass theorem for restriction. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremrelres 4728 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 4729 Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25. (Contributed by NM, 9-Aug-1994.)

Theoremresabs2 4730 Absorption law for restriction. (Contributed by NM, 27-Mar-1998.)

Theoremresidm 4731 Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.)

Theoremresima 4732 A restriction to an image. (Contributed by NM, 29-Sep-2004.)

Theoremresima2 4733 Image under a restricted class. (Contributed by FL, 31-Aug-2009.)

Theoremxpssres 4734 Restriction of a constant function (or other cross product). (Contributed by Stefan O'Rear, 24-Jan-2015.)

Theoremelres 4735* Membership in a restriction. (Contributed by Scott Fenton, 17-Mar-2011.)

Theoremelsnres 4736* Memebership in restriction to a singleton. (Contributed by Scott Fenton, 17-Mar-2011.)

Theoremrelssres 4737 Simplification law for restriction. (Contributed by NM, 16-Aug-1994.)

Theoremresdm 4738 A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.)

Theoremresexg 4739 The restriction of a set is a set. (Contributed by NM, 28-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremresex 4740 The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.)

Theoremresindm 4741 When restricting a relation, intersecting with the domain of the relation has no effect. (Contributed by FL, 6-Oct-2008.)

Theoremresdmdfsn 4742 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 4743* Restriction of a class abstraction of ordered pairs. (Contributed by NM, 5-Nov-2002.)

Theoremresiexg 4744 The existence of a restricted identity function, proved without using the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.)

Theoremiss 4745 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 4746* Restriction of a class abstraction of ordered pairs. (Contributed by NM, 24-Aug-2007.)

Theoremresmpt 4747* Restriction of the mapping operation. (Contributed by Mario Carneiro, 15-Jul-2013.)

Theoremresmpt3 4748* Unconditional restriction of the mapping operation. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Proof shortened by Mario Carneiro, 22-Mar-2015.)

Theoremresmptf 4749 Restriction of the mapping operation. (Contributed by Thierry Arnoux, 28-Mar-2017.)

Theoremresmptd 4750* Restriction of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremdfres2 4751* Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.)

Theoremopabresid 4752* The restricted identity expressed with the class builder. (Contributed by FL, 25-Apr-2012.)

Theoremmptresid 4753* The restricted identity expressed with the maps-to notation. (Contributed by FL, 25-Apr-2012.)

Theoremdmresi 4754 The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.)

Theoremresid 4755 Any relation restricted to the universe is itself. (Contributed by NM, 16-Mar-2004.)

Theoremimaeq1 4756 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Theoremimaeq2 4757 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Theoremimaeq1i 4758 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)

Theoremimaeq2i 4759 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)

Theoremimaeq1d 4760 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)

Theoremimaeq2d 4761 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)

Theoremimaeq12d 4762 Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.)

Theoremdfima2 4763* 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 4764* 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 4765* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 20-Jan-2007.)

Theoremelima 4766* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 19-Apr-2004.)

Theoremelima2 4767* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 11-Aug-2004.)

Theoremelima3 4768* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 14-Aug-1994.)

Theoremnfima 4769 Bound-variable hypothesis builder for image. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremnfimad 4770 Deduction version of bound-variable hypothesis builder nfima 4769. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremimadmrn 4771 The image of the domain of a class is the range of the class. (Contributed by NM, 14-Aug-1994.)

Theoremimassrn 4772 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 4773 The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by NM, 24-Jul-1995.)

Theoremimaex 4774 The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by JJ, 24-Sep-2021.)

Theoremimai 4775 Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.)

Theoremrnresi 4776 The range of the restricted identity function. (Contributed by NM, 27-Aug-2004.)

Theoremresiima 4777 The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)

Theoremima0 4778 Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.)

Theorem0ima 4779 Image under the empty relation. (Contributed by FL, 11-Jan-2007.)

Theoremcsbima12g 4780 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 4781 A class whose image under another is empty is disjoint with the other's domain. (Contributed by FL, 24-Jan-2007.)

Theoremcnvimass 4782 A preimage under any class is included in the domain of the class. (Contributed by FL, 29-Jan-2007.)

Theoremcnvimarndm 4783 The preimage of the range of a class is the domain of the class. (Contributed by Jeff Hankins, 15-Jul-2009.)

Theoremimasng 4784* The image of a singleton. (Contributed by NM, 8-May-2005.)

Theoremelreimasng 4785 Elementhood in the image of a singleton. (Contributed by Jim Kingdon, 10-Dec-2018.)

Theoremelimasn 4786 Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremelimasng 4787 Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.)

Theoremargs 4788* 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 4789 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 4790 Any set is equal to its preimage under the converse epsilon relation. (Contributed by Mario Carneiro, 9-Mar-2013.)

Theoreminiseg 4791* 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 4792* Alternate definition of set-like relation. (Contributed by Mario Carneiro, 23-Jun-2015.)
Se

Theoremexse2 4793 Any set relation is set-like. (Contributed by Mario Carneiro, 22-Jun-2015.)
Se

Theoremimass1 4794 Subset theorem for image. (Contributed by NM, 16-Mar-2004.)

Theoremimass2 4795 Subset theorem for image. Exercise 22(a) of [Enderton] p. 53. (Contributed by NM, 22-Mar-1998.)

Theoremndmima 4796 The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998.)

Theoremrelcnv 4797 A converse is a relation. Theorem 12 of [Suppes] p. 62. (Contributed by NM, 29-Oct-1996.)

Theoremrelbrcnvg 4798 When is a relation, the sethood assumptions on brcnv 4607 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)

Theoremrelbrcnv 4799 When is a relation, the sethood assumptions on brcnv 4607 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)

Theoremcotr 4800* 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.)

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