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

Theoremelima 4701* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 19-Apr-2004.)

Theoremelima2 4702* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 11-Aug-2004.)

Theoremelima3 4703* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 14-Aug-1994.)

Theoremnfima 4704 Bound-variable hypothesis builder for image. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremnfimad 4705 Deduction version of bound-variable hypothesis builder nfima 4704. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremimadmrn 4706 The image of the domain of a class is the range of the class. (Contributed by NM, 14-Aug-1994.)

Theoremimassrn 4707 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 4708 The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by NM, 24-Jul-1995.)

Theoremimai 4709 Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.)

Theoremrnresi 4710 The range of the restricted identity function. (Contributed by NM, 27-Aug-2004.)

Theoremresiima 4711 The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)

Theoremima0 4712 Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.)

Theorem0ima 4713 Image under the empty relation. (Contributed by FL, 11-Jan-2007.)

Theoremcsbima12g 4714 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 4715 A class whose image under another is empty is disjoint with the other's domain. (Contributed by FL, 24-Jan-2007.)

Theoremcnvimass 4716 A preimage under any class is included in the domain of the class. (Contributed by FL, 29-Jan-2007.)

Theoremcnvimarndm 4717 The preimage of the range of a class is the domain of the class. (Contributed by Jeff Hankins, 15-Jul-2009.)

Theoremimasng 4718* The image of a singleton. (Contributed by NM, 8-May-2005.)

Theoremelreimasng 4719 Elementhood in the image of a singleton. (Contributed by Jim Kingdon, 10-Dec-2018.)

Theoremelimasn 4720 Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremelimasng 4721 Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.)

Theoremargs 4722* 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 4723 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 4724 Any set is equal to its preimage under the converse epsilon relation. (Contributed by Mario Carneiro, 9-Mar-2013.)

Theoreminiseg 4725* 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 4726* Alternate definition of set-like relation. (Contributed by Mario Carneiro, 23-Jun-2015.)
Se

Theoremexse2 4727 Any set relation is set-like. (Contributed by Mario Carneiro, 22-Jun-2015.)
Se

Theoremimass1 4728 Subset theorem for image. (Contributed by NM, 16-Mar-2004.)

Theoremimass2 4729 Subset theorem for image. Exercise 22(a) of [Enderton] p. 53. (Contributed by NM, 22-Mar-1998.)

Theoremndmima 4730 The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998.)

Theoremrelcnv 4731 A converse is a relation. Theorem 12 of [Suppes] p. 62. (Contributed by NM, 29-Oct-1996.)

Theoremrelbrcnvg 4732 When is a relation, the sethood assumptions on brcnv 4546 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)

Theoremrelbrcnv 4733 When is a relation, the sethood assumptions on brcnv 4546 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)

Theoremcotr 4734* 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 4735* Two ways to state a relation is reflexive. Adapted from Tarski. (Contributed by FL, 15-Jan-2012.) (Revised by NM, 30-Mar-2016.)

Theoremcnvsym 4736* 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 4737* 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 4738* Two ways of saying a relation is antisymmetric and reflexive. is the field of a relation by relfld 4874. (Contributed by NM, 6-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremintirr 4739* 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 4740* Two ways of saying that two elements have an upper bound. (Contributed by Mario Carneiro, 3-Nov-2015.)

Theoremcodir 4741* Two ways of saying a relation is directed. (Contributed by Mario Carneiro, 22-Nov-2013.)

Theoremqfto 4742* A quantifier-free way of expressing the total order predicate. (Contributed by Mario Carneiro, 22-Nov-2013.)

Theoremxpidtr 4743 A square cross product is a transitive relation. (Contributed by FL, 31-Jul-2009.)

Theoremtrin2 4744 The intersection of two transitive classes is transitive. (Contributed by FL, 31-Jul-2009.)

Theorempoirr2 4745 A partial order relation is irreflexive. (Contributed by Mario Carneiro, 2-Nov-2015.)

Theoremtrinxp 4746 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 4747 A strict order relation is irreflexive. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)

Theoremsotri 4748 A strict order relation is a transitive relation. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)

Theoremson2lpi 4749 A strict order relation has no 2-cycle loops. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)

Theoremsotri2 4750 A transitivity relation. (Read B < A and B < C implies A < C .) (Contributed by Mario Carneiro, 10-May-2013.)

Theoremsotri3 4751 A transitivity relation. (Read A < B and C < B implies A < C .) (Contributed by Mario Carneiro, 10-May-2013.)

Theorempoleloe 4752 Express "less than or equals" for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theorempoltletr 4753 Transitive law for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremcnvopab 4754* The converse of a class abstraction of ordered pairs. (Contributed by NM, 11-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcnv0 4755 The converse of the empty set. (Contributed by NM, 6-Apr-1998.)

Theoremcnvi 4756 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 4757 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 4758 Distributive law for converse over set difference. (Contributed by Mario Carneiro, 26-Jun-2014.)

Theoremcnvin 4759 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 4760 Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.)

Theoremrnin 4761 The range of an intersection belongs the intersection of ranges. Theorem 9 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)

Theoremrniun 4762 The range of an indexed union. (Contributed by Mario Carneiro, 29-May-2015.)

Theoremrnuni 4763* 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 4764 Distributive law for image over union. Theorem 35 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)

Theoremimaundir 4765 The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)

Theoremdminss 4766 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 4767 An upper bound for intersection with an image. Theorem 41 of [Suppes] p. 66. (Contributed by NM, 11-Aug-2004.)

Theoreminimass 4768 The image of an intersection (Contributed by Thierry Arnoux, 16-Dec-2017.)

Theoreminimasn 4769 The intersection of the image of singleton (Contributed by Thierry Arnoux, 16-Dec-2017.)

Theoremcnvxp 4770 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 4771 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 4772* The cross product of inhabited classes is inhabited. (Contributed by Jim Kingdon, 11-Dec-2018.)

Theoremxpm 4773* The cross product of inhabited classes is inhabited. (Contributed by Jim Kingdon, 13-Dec-2018.)

Theoremxpeq0r 4774 A cross product is empty if at least one member is empty. (Contributed by Jim Kingdon, 12-Dec-2018.)

Theoremxpdisj1 4775 Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)

Theoremxpdisj2 4776 Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)

Theoremxpsndisj 4777 Cross products with two different singletons are disjoint. (Contributed by NM, 28-Jul-2004.)

Theoremdjudisj 4778* Disjoint unions with disjoint index sets are disjoint. (Contributed by Stefan O'Rear, 21-Nov-2014.)

Theoremresdisj 4779 A double restriction to disjoint classes is the empty set. (Contributed by NM, 7-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremrnxpm 4780* The range of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37, with non-empty changed to inhabited. (Contributed by Jim Kingdon, 12-Dec-2018.)

Theoremdmxpss 4781 The domain of a cross product is a subclass of the first factor. (Contributed by NM, 19-Mar-2007.)

Theoremrnxpss 4782 The range of a cross product is a subclass of the second factor. (Contributed by NM, 16-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremrnxpid 4783 The range of a square cross product. (Contributed by FL, 17-May-2010.)

Theoremssxpbm 4784* A cross-product subclass relationship is equivalent to the relationship for its components. (Contributed by Jim Kingdon, 12-Dec-2018.)

Theoremssxp1 4785* Cross product subset cancellation. (Contributed by Jim Kingdon, 14-Dec-2018.)

Theoremssxp2 4786* Cross product subset cancellation. (Contributed by Jim Kingdon, 14-Dec-2018.)

Theoremxp11m 4787* The cross product of inhabited classes is one-to-one. (Contributed by Jim Kingdon, 13-Dec-2018.)

Theoremxpcanm 4788* Cancellation law for cross-product. (Contributed by Jim Kingdon, 14-Dec-2018.)

Theoremxpcan2m 4789* Cancellation law for cross-product. (Contributed by Jim Kingdon, 14-Dec-2018.)

Theoremxpexr2m 4790* If a nonempty cross product is a set, so are both of its components. (Contributed by Jim Kingdon, 14-Dec-2018.)

Theoremssrnres 4791 Subset of the range of a restriction. (Contributed by NM, 16-Jan-2006.)

Theoremrninxp 4792* Range of the intersection with a cross product. (Contributed by NM, 17-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremdminxp 4793* Domain of the intersection with a cross product. (Contributed by NM, 17-Jan-2006.)

Theoremimainrect 4794 Image of a relation restricted to a rectangular region. (Contributed by Stefan O'Rear, 19-Feb-2015.)

Theoremxpima1 4795 The image by a cross product. (Contributed by Thierry Arnoux, 16-Dec-2017.)

Theoremxpima2m 4796* The image by a cross product. (Contributed by Thierry Arnoux, 16-Dec-2017.)

Theoremxpimasn 4797 The image of a singleton by a cross product. (Contributed by Thierry Arnoux, 14-Jan-2018.)

Theoremcnvcnv3 4798* The set of all ordered pairs in a class is the same as the double converse. (Contributed by Mario Carneiro, 16-Aug-2015.)

Theoremdfrel2 4799 Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 29-Dec-1996.)

Theoremdfrel4v 4800* A relation can be expressed as the set of ordered pairs in it. (Contributed by Mario Carneiro, 16-Aug-2015.)

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