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Theorem List for Metamath Proof Explorer - 5101-5200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcnvi 5101 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 5102 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 5103 Distributive law for converse over set difference. (Contributed by Mario Carneiro, 26-Jun-2014.)

Theoremcnvin 5104 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 5105 Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.)

Theoremrnin 5106 The range of an intersection belongs the intersection of ranges. Theorem 9 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)

Theoremrniun 5107 The range of an indexed union. (Contributed by Mario Carneiro, 29-May-2015.)

Theoremrnuni 5108* 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 5109 Distributive law for image over union. Theorem 35 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)

Theoremimaundir 5110 The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)

Theoremdminss 5111 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 5112 An upper bound for intersection with an image. Theorem 41 of [Suppes] p. 66. (Contributed by NM, 11-Aug-2004.)

Theoremcnvxp 5113 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 5114 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.)

Theoremxpnz 5115 The cross product of nonempty classes is nonempty. (Variation of a theorem contributed by Raph Levien, 30-Jun-2006.) (Contributed by NM, 30-Jun-2006.)

Theoremxpeq0 5116 At least one member of an empty cross product is empty. (Contributed by NM, 27-Aug-2006.)

Theoremxpdisj1 5117 Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)

Theoremxpdisj2 5118 Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)

Theoremxpsndisj 5119 Cross products with two different singletons are disjoint. (Contributed by NM, 28-Jul-2004.)

Theoremdjudisj 5120* Disjoint unions with disjoint index sets are disjoint. (Contributed by Stefan O'Rear, 21-Nov-2014.)

Theoremresdisj 5121 A double restriction to disjoint classes is the empty set. (Contributed by NM, 7-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremrnxp 5122 The range of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.)

Theoremdmxpss 5123 The domain of a cross product is a subclass of the first factor. (Contributed by NM, 19-Mar-2007.)

Theoremrnxpss 5124 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 5125 The range of a square cross product. (Contributed by FL, 17-May-2010.)

Theoremssxpb 5126 A cross-product subclass relationship is equivalent to the relationship for it components. (Contributed by NM, 17-Dec-2008.)

Theoremxp11 5127 The cross product of non-empty classes is one-to-one. (Contributed by NM, 31-May-2008.)

Theoremxpcan 5128 Cancellation law for cross-product. (Contributed by NM, 30-Aug-2011.)

Theoremxpcan2 5129 Cancellation law for cross-product. (Contributed by NM, 30-Aug-2011.)

Theoremxpexr 5130 If a cross product is a set, one of its components must be a set. (Contributed by NM, 27-Aug-2006.)

Theoremxpexr2 5131 If a nonempty cross product is a set, so are both of its components. (Contributed by NM, 27-Aug-2006.)

Theoremssrnres 5132 Subset of the range of a restriction. (Contributed by NM, 16-Jan-2006.)

Theoremrninxp 5133* Range of the intersection with a cross product. (Contributed by NM, 17-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremdminxp 5134* Domain of the intersection with a cross product. (Contributed by NM, 17-Jan-2006.)

Theoremimainrect 5135 Image of a relation restricted to a rectangular region. (Contributed by Stefan O'Rear, 19-Feb-2015.)

Theoremsossfld 5136 The base set of a strict order is contained in the field of the relation, except possibly for one element (note that ). (Contributed by Mario Carneiro, 27-Apr-2015.)

Theoremsofld 5137 The base set of a nonempty strict order is the same as the field of the relation. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremsoex 5138 If the relation in a strict order is a set, then the base field is also a set. (Contributed by Mario Carneiro, 27-Apr-2015.)

Theoremcnvcnv3 5139* The set of all ordered pairs in a class is the same as the double converse. (Contributed by Mario Carneiro, 16-Aug-2015.)

Theoremdfrel2 5140 Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 29-Dec-1996.)

Theoremdfrel4v 5141* A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 5584 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.)

Theoremcnvcnv 5142 The double converse of a class strips out all elements that are not ordered pairs. (Contributed by NM, 8-Dec-2003.)

Theoremcnvcnv2 5143 The double converse of a class equals its restriction to the universe. (Contributed by NM, 8-Oct-2007.)

Theoremcnvcnvss 5144 The double converse of a class is a subclass. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 23-Jul-2004.)

Theoremcnveqb 5145 Equality theorem for converse. (Contributed by FL, 19-Sep-2011.)

Theoremcnveq0 5146 A relation empty iff its converse is empty. (Contributed by FL, 19-Sep-2011.)

Theoremdfrel3 5147 Alternate definition of relation. (Contributed by NM, 14-May-2008.)

Theoremdmresv 5148 The domain of a universal restriction. (Contributed by NM, 14-May-2008.)

Theoremrnresv 5149 The range of a universal restriction. (Contributed by NM, 14-May-2008.)

Theoremdfrn4 5150 Range defined in terms of image. (Contributed by NM, 14-May-2008.)

Theoremrescnvcnv 5151 The restriction of the double converse of a class. (Contributed by NM, 8-Apr-2007.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcnvcnvres 5152 The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.)

Theoremimacnvcnv 5153 The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.)

Theoremdmsnn0 5154 The domain of a singleton is nonzero iff the singleton argument is an ordered pair. (Contributed by NM, 14-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremrnsnn0 5155 The range of a singleton is nonzero iff the singleton argument is an ordered pair. (Contributed by NM, 14-Dec-2008.)

Theoremdmsn0 5156 The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.)

Theoremcnvsn0 5157 The converse of the singleton of the empty set is empty. (Contributed by Mario Carneiro, 30-Aug-2015.)

Theoremdmsn0el 5158 The domain of a singleton is empty if the singleton's argument contains the empty set. (Contributed by NM, 15-Dec-2008.)

Theoremrelsn2 5159 A singleton is a relation iff it has a nonempty domain. (Contributed by NM, 25-Sep-2013.)

Theoremdmsnopg 5160 The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremdmsnopss 5161 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 5162 The domain of an unordered pair of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremdmsnop 5163 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 5164 The domain of an unordered pair of ordered pairs. (Contributed by NM, 13-Sep-2011.)

Theoremdmtpop 5165 The domain of an unordered triple of ordered pairs. (Contributed by NM, 14-Sep-2011.)

Theoremcnvcnvsn 5166 Double converse of a singleton of an ordered pair. (Unlike cnvsn 5171, this does not need any sethood assumptions on and .) (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremdmsnsnsn 5167 The domain of the singleton of the singleton of a singleton. (Contributed by NM, 15-Sep-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremrnsnopg 5168 The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremrnsnop 5169 The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremop1sta 5170 Extract the first member of an ordered pair. (See op2nda 5173 to extract the second member, op1stb 4585 for an alternate version, and op1st 6144 for the preferred version.) (Contributed by Raph Levien, 4-Dec-2003.)

Theoremcnvsn 5171 Converse of a singleton of an ordered pair. (Contributed by NM, 11-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremop2ndb 5172 Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4585 to extract the first member, op2nda 5173 for an alternate version, and op2nd 6145 for the preferred version.) (Contributed by NM, 25-Nov-2003.)

Theoremop2nda 5173 Extract the second member of an ordered pair. (See op1sta 5170 to extract the first member, op2ndb 5172 for an alternate version, and op2nd 6145 for the preferred version.) (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcnvsng 5174 Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.)

Theoremopswap 5175 Swap the members of an ordered pair. (Contributed by NM, 14-Dec-2008.) (Revised by Mario Carneiro, 30-Aug-2015.)

Theoremelxp4 5176 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 5177, elxp6 6167, and elxp7 6168. (Contributed by NM, 17-Feb-2004.)

Theoremelxp5 5177 Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 5176 when the double intersection does not create class existence problems (caused by int0 3892). (Contributed by NM, 1-Aug-2004.)

Theoremcnvresima 5178 An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.)

Theoremresdm2 5179 A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.)

Theoremresdmres 5180 Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.)

Theoremimadmres 5181 The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.)

Theoremmptpreima 5182* The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Theoremmptiniseg 5183* Converse singleton image of a function defined by maps-to. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Theoremdmmpt 5184 The domain of the mapping operation in general. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 22-Mar-2015.)

Theoremdmmptss 5185* The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.)

Theoremdmmptg 5186* The domain of the mapping operation is the stated domain, if the function value is always a set. (Contributed by Mario Carneiro, 9-Feb-2013.) (Revised by Mario Carneiro, 14-Sep-2013.)

Theoremrelco 5187 A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.)

Theoremdfco2 5188* Alternate definition of a class composition, using only one bound variable. (Contributed by NM, 19-Dec-2008.)

Theoremdfco2a 5189* Generalization of dfco2 5188, where can have any value between and . (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcoundi 5190 Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcoundir 5191 Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcores 5192 Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremresco 5193 Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)

Theoremimaco 5194 Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.)

Theoremrnco 5195 The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.)

Theoremrnco2 5196 The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.)

Theoremdmco 5197 The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.)

Theoremcoiun 5198* Composition with an indexed union. (Contributed by NM, 21-Dec-2008.)

Theoremcocnvcnv1 5199 A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.)

Theoremcocnvcnv2 5200 A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.)

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