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Theorem List for New Foundations Explorer - 5001-5100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremresopab2 5001* Restriction of a class abstraction of ordered pairs. (Contributed by set.mm contributors, 24-Aug-2007.)

Theoremdfres2 5002* Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.)

Theoremopabresid 5003* The restricted identity expressed with the class builder. (Contributed by FL, 25-Apr-2012.)

Theoremdmresi 5004 The domain of a restricted identity function. (Contributed by set.mm contributors, 27-Aug-2004.)

Theoremresid 5005 Any class restricted to the universe is itself. (Contributed by set.mm contributors, 16-Mar-2004.) (Revised by Scott Fenton, 18-Apr-2021.)

Theoremresima 5006 A restriction to an image. (Contributed by set.mm contributors, 29-Sep-2004.)

Theoremresima2 5007 Image under a restricted class. (Contributed by FL, 31-Aug-2009.)

Theoremimadmrn 5008 The image of the domain of a class is the range of the class. (Contributed by set.mm contributors, 14-Aug-1994.)

Theoremimassrn 5009 The image of a class is a subset of its range. Theorem 3.16(xi) of [Monk1] p. 39. (Contributed by set.mm contributors, 31-Mar-1995.)

Theoremimai 5010 Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by set.mm contributors, 30-Apr-1998.)

Theoremrnresi 5011 The range of the restricted identity function. (Contributed by set.mm contributors, 27-Aug-2004.)

Theoremresiima 5012 The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)

Theoremima0 5013 Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by set.mm contributors, 20-May-1998.)

Theorem0ima 5014 Image under the empty relation. (Contributed by FL, 11-Jan-2007.)

Theoremimadisj 5015 A class whose image under another is empty is disjoint with the other's domain. (Contributed by FL, 24-Jan-2007.)

Theoremcnvimass 5016 A preimage under any class is included in the domain of the class. (Contributed by FL, 29-Jan-2007.)

Theoremcnvimarndm 5017 The preimage of the range of a class is the domain of the class. (Contributed by Jeff Hankins, 15-Jul-2009.)

Theoremimasn 5018* The image of a singleton. (Contributed by set.mm contributors, 9-Jan-2015.)

Theoremelimasn 5019 Membership in an image of a singleton. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 15-Mar-2004.) (Revised by set.mm contributors, 27-Aug-2011.)

Theoremeliniseg 5020 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. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 28-Apr-2004.) (Revised by set.mm contributors, 27-Aug-2011.)

Theoremepini 5021 Any set is equal to its preimage under the converse epsilon relation. (Contributed by Mario Carneiro, 9-Mar-2013.)

Theoreminiseg 5022* An idiom that signifies an initial segment of an ordering, used, for example, in Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by set.mm contributors, 28-Apr-2004.)

Theoremimass1 5023 Subset theorem for image. (Contributed by set.mm contributors, 16-Mar-2004.)

Theoremimass2 5024 Subset theorem for image. Exercise 22(a) of [Enderton] p. 53. (Contributed by set.mm contributors, 22-Mar-1998.)

Theoremndmima 5025 The image of a singleton outside the domain is empty. (Contributed by set.mm contributors, 22-May-1998.)

Theoremcotr 5026* Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 27-Dec-1996.) (Revised by set.mm contributors, 27-Aug-2011.)

Theoremcnvsym 5027* Two ways of saying a relation is symmetric. Similar to definition of symmetry in [Schechter] p. 51. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 28-Dec-1996.) (Revised by set.mm contributors, 27-Aug-2011.)

Theoremintasym 5028* Two ways of saying a relation is antisymmetric. Definition of antisymmetry in [Schechter] p. 51. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 9-Sep-2004.) (Revised by set.mm contributors, 27-Aug-2011.)

Theoremintirr 5029* Two ways of saying a relation is irreflexive. Definition of irreflexivity in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Revised by Andrew Salmon, 27-Aug-2011.)

Theoremcnvopab 5030* The converse of a class abstraction of ordered pairs. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 11-Dec-2003.) (Revised by set.mm contributors, 27-Aug-2011.)

Theoremcnv0 5031 The converse of the empty set. (Contributed by set.mm contributors, 6-Apr-1998.)

Theoremcnvi 5032 The converse of the identity relation. Theorem 3.7(ii) of [Monk1] p. 36. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 26-Apr-1998.) (Revised by set.mm contributors, 27-Aug-2011.)

Theoremcnvun 5033 The converse of a union is the union of converses. Theorem 16 of [Suppes] p. 62. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 25-Mar-1998.) (Revised by set.mm contributors, 27-Aug-2011.)

Theoremcnvdif 5034 Distributive law for converse over set difference. (Contributed by set.mm contributors, 26-Jun-2014.)

Theoremcnvin 5035 Distributive law for converse over intersection. Theorem 15 of [Suppes] p. 62. (Contributed by set.mm contributors, 25-Mar-1998.) (Revised by set.mm contributors, 26-Jun-2014.)

Theoremrnun 5036 Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by set.mm contributors, 24-Mar-1998.)

Theoremrnin 5037 The range of an intersection belongs the intersection of ranges. Theorem 9 of [Suppes] p. 60. (Contributed by set.mm contributors, 15-Sep-2004.)

Theoremrnuni 5038* The range of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by set.mm contributors, 17-Mar-2004.)

Theoremimaundi 5039 Distributive law for image over union. Theorem 35 of [Suppes] p. 65. (Contributed by set.mm contributors, 30-Sep-2002.)

Theoremimaundir 5040 The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)

Theoremdminss 5041 An upper bound for intersection with a domain. Theorem 40 of [Suppes] p. 66, who calls it "somewhat surprising." (Contributed by set.mm contributors, 11-Aug-2004.)

Theoremimainss 5042 An upper bound for intersection with an image. Theorem 41 of [Suppes] p. 66. (Contributed by set.mm contributors, 11-Aug-2004.)

Theoremcnvxp 5043 The converse of a cross product. Exercise 11 of [Suppes] p. 67. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 14-Aug-1999.) (Revised by set.mm contributors, 27-Aug-2011.)

Theoremxp0 5044 The cross product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by set.mm contributors, 12-Apr-2004.)

Theoremxpnz 5045 The cross product of nonempty classes is nonempty. (Variation of a theorem contributed by Raph Levien, 30-Jun-2006.) (Contributed by set.mm contributors, 30-Jun-2006.) (Revised by set.mm contributors, 19-Apr-2007.)

Theoremxpeq0 5046 At least one member of an empty cross product is empty. (Contributed by set.mm contributors, 27-Aug-2006.)

Theoremxpdisj1 5047 Cross products with disjoint sets are disjoint. (Contributed by set.mm contributors, 13-Sep-2004.)

Theoremxpdisj2 5048 Cross products with disjoint sets are disjoint. (Contributed by set.mm contributors, 13-Sep-2004.)

Theoremxpsndisj 5049 Cross products with two different singletons are disjoint. (Contributed by set.mm contributors, 28-Jul-2004.) (Revised by set.mm contributors, 3-Jun-2007.)

Theoremresdisj 5050 A double restriction to disjoint classes is the empty set. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 7-Oct-2004.) (Revised by set.mm contributors, 27-Aug-2011.)

Theoremrnxp 5051 The range of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by set.mm contributors, 12-Apr-2004.) (Revised by set.mm contributors, 9-Apr-2007.)

Theoremdmxpss 5052 The domain of a cross product is a subclass of the first factor. (Contributed by set.mm contributors, 19-Mar-2007.)

Theoremrnxpss 5053 The range of a cross product is a subclass of the second factor. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 16-Jan-2006.) (Revised by set.mm contributors, 27-Aug-2011.)

Theoremrnxpid 5054 The range of a square cross product. (Contributed by FL, 17-May-2010.)

Theoremssxpb 5055 A cross-product subclass relationship is equivalent to the relationship for it components. (Contributed by set.mm contributors, 17-Dec-2008.)

Theoremxp11 5056 The cross product of non-empty classes is one-to-one. (Contributed by set.mm contributors, 31-May-2008.)

Theoremxpcan 5057 Cancellation law for cross-product. (Contributed by set.mm contributors, 30-Aug-2011.)

Theoremxpcan2 5058 Cancellation law for cross-product. (Contributed by set.mm contributors, 30-Aug-2011.)

Theoremssrnres 5059 Subset of the range of a restriction. (Contributed by set.mm contributors, 16-Jan-2006.)

Theoremrninxp 5060* Range of the intersection with a cross product. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 17-Jan-2006.) (Revised by set.mm contributors, 27-Aug-2011.)

Theoremdminxp 5061* Domain of the intersection with a cross product. (Contributed by set.mm contributors, 17-Jan-2006.)

Theoremcnvcnv 5062 The double converse of a class is the original class. (Contributed by Scott Fenton, 17-Apr-2021.)

Theoremcnveqb 5063 Equality theorem for converse. (Contributed by FL, 19-Sep-2011.) (Revised by Scott Fenton, 17-Apr-2021.)

Theoremdmsnn0 5064 The domain of a singleton is nonzero iff the singleton argument is a set. (Contributed by NM, 14-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Scott Fenton, 19-Apr-2021.)

Theoremrnsnn0 5065 The range of a singleton is nonzero iff the singleton argument is a set. (Contributed by set.mm contributors, 14-Dec-2008.) (Revised by Scott Fenton, 19-Apr-2021.)

Theoremdmsnopg 5066 The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremdmsnopss 5067 The domain of a singleton of an ordered pair is a subset of the singleton of the first member (with no sethood assumptions on ). (Contributed by Mario Carneiro, 30-Apr-2015.)

Theoremdmpropg 5068 The domain of an unordered pair of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremdmsnop 5069 The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremdmprop 5070 The domain of an unordered pair of ordered pairs. (Contributed by NM, 13-Sep-2011.)

Theoremdmtpop 5071 The domain of an unordered triple of ordered pairs. (Contributed by NM, 14-Sep-2011.)

Theoremop1sta 5072 Extract the first member of an ordered pair. (Contributed by Raph Levien, 4-Dec-2003.)

Theoremcnvsn 5073 Converse of a singleton of an ordered pair. (Contributed by NM, 11-May-1998.)

Theoremopswap 5074 Swap the members of an ordered pair. (Contributed by set.mm contributors, 14-Dec-2008.)

Theoremrnsnop 5075 The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by set.mm contributors, 24-Jul-2004.)

Theoremop2nda 5076 Extract the second member of an ordered pair. (Contributed by set.mm contributors, 9-Jan-2015.)

Theoremcnvresima 5077 An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.)

Theoremresdmres 5078 Restriction to the domain of a restriction. (Contributed by set.mm contributors, 8-Apr-2007.)

Theoremimadmres 5079 The image of the domain of a restriction. (Contributed by set.mm contributors, 8-Apr-2007.)

Theoremdfco2 5080* Alternate definition of a class composition, using only one bound variable. (Contributed by set.mm contributors, 19-Dec-2008.)

Theoremdfco2a 5081* Generalization of dfco2 5080, where can have any value between and . (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 21-Dec-2008.) (Revised by set.mm contributors, 27-Aug-2011.)

Theoremcoundi 5082 Class composition distributes over union. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 21-Dec-2008.) (Revised by set.mm contributors, 27-Aug-2011.)

Theoremcoundir 5083 Class composition distributes over union. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 21-Dec-2008.) (Revised by set.mm contributors, 27-Aug-2011.)

Theoremcores 5084 Restricted first member of a class composition. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 12-Oct-2004.) (Revised by set.mm contributors, 27-Aug-2011.)

Theoremresco 5085 Associative law for the restriction of a composition. (Contributed by set.mm contributors, 12-Dec-2006.)

Theoremimaco 5086 Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.)

Theoremrnco 5087 The range of the composition of two classes. (Contributed by set.mm contributors, 12-Dec-2006.)

Theoremrnco2 5088 The range of the composition of two classes. (Contributed by set.mm contributors, 27-Mar-2008.)

Theoremdmco 5089 The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by set.mm contributors, 4-Feb-2004.)

Theoremcoiun 5090* Composition with an indexed union. (Contributed by set.mm contributors, 21-Dec-2008.)

Theoremcores2 5091 Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by set.mm contributors, 11-Dec-2006.)

Theoremco02 5092 Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by set.mm contributors, 24-Apr-2004.)

Theoremco01 5093 Composition with the empty set. (Contributed by set.mm contributors, 24-Apr-2004.)

Theoremcoi1 5094 Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by set.mm contributors, 22-Apr-2004.) (Revised by Scott Fenton, 14-Apr-2021.)

Theoremcoi2 5095 Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by set.mm contributors, 22-Apr-2004.) (Revised by Scott Fenton, 17-Apr-2021.)

Theoremcoires1 5096 Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Scott Fenton, 17-Apr-2021.)

Theoremcoass 5097 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 set.mm contributors, 27-Jan-1997.)

Theoremcnvtr 5098 A class is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.) (Revised by Scott Fenton, 18-Apr-2021.)

Theoremssdmrn 5099 A class is included in the cross product of its domain and range. Exercise 4.12(t) of [Mendelson] p. 235. (Contributed by set.mm contributors, 3-Aug-1994.) (Revised by Scott Fenton, 15-Apr-2021.)

Theoremdfcnv2 5100 Definition of converse in terms of image and Swap . (Contributed by set.mm contributors, 8-Jan-2015.)
Swap

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