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Statement List for Metamath Proof Explorer - 4101-4200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeqrelrel 4101* Extensionality principle for ordered triples (used by 2-place operations df-oprab 4927), analogous to eqrel 4092. Use relrelss 4428 to express the antecedent in terms of the relation predicate.
 
Theoremelrel 4102* A member of a relation is an ordered pair.
 
Theoremrelsn 4103 A singleton is a relation iff it is an ordered pair.
   =>   
 
Theoremrelsnop 4104 A singleton of an ordered pair is a relation.
   =>   
 
Theoremxpss12 4105 Subset theorem for cross product. Generalization of Theorem 101 of [Suppes] p. 52. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
 
Theoremxpss 4106 A cross product is included in the ordered pair universe. Exercise 3 of [TakeutiZaring] p. 25.
 
Theoremrelxp 4107 A cross product is a relation. Theorem 3.13(i) of [Monk1] p. 37.
 
Theoremxpss1 4108 Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)
 
Theoremxpss2 4109 Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)
 
Theoremxpsspw 4110 A cross product is included in the power of the power of the union of its arguments.
 
TheoremxpsspwOLD 4111 A cross product is included in the power of the power of the union of its arguments.
 
Theoremunixpss 4112 The double class union of a cross product is included in the union of its arguments.
 
Theoremxpexg 4113 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23.
 
Theoremxpex 4114 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23.
   &       =>   
 
Theoremrelun 4115 The union of two relations is a relation. Compare Exercise 5 of [TakeutiZaring] p. 25.
 
Theoremrelin1 4116 The intersection with a relation is a relation.
 
Theoremrelin2 4117 The intersection with a relation is a relation.
 
Theoremreldif 4118 A difference cutting down a relation is a relation.
 
Theoremreliun 4119 An indexed union is a relation iff each member of its indexed family is a relation.
 
Theoremreliin 4120 An indexed intersection is a relation if if at least one of the member of the indexed family is a relation.
 
Theoremreluni 4121* The union of a class is a relation iff any member is a relation. Exercise 6 of [TakeutiZaring] p. 25 and its converse.
 
Theoremrelint 4122* The intersection of a class is a relation if at least one member is a relation.
 
Theoremrel0 4123 The empty set is a relation.
 
Theoremrelopabi 4124 A class of ordered pairs is a relation. (The proof was shortened by Mario Carneiro, 21-Dec-2013.)
   =>   
 
Theoremrelopab 4125 A class of ordered pairs is a relation. (Unnecessary distinct variable restrictions were removed by Alan Sare, 9-Jul-2013.) (The proof was shortened by Mario Carneiro, 21-Dec-2013.)
 
TheoremrelopabOLD 4126* Obsolete proof of relopab 4125 as of 21-Dec-2013.
 
Theoremreli 4127 The identity relation is a relation. Part of Exercise 4.12(p) of [Mendelson] p. 235.
 
Theoremrele 4128 The membership relation is a relation.
 
Theoremopabid2 4129* A relation expressed as an ordered pair abstraction.
 
Theoreminopab 4130* Intersection of two ordered pair class abstractions.
 
Theoreminxp 4131 The intersection of two cross products. Exercise 9 of [TakeutiZaring] p. 25. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
 
Theoremxpindi 4132 Distributive law for cross product over intersection. Theorem 102 of [Suppes] p. 52.
 
Theoremxpindir 4133 Distributive law for cross product over intersection. Similar to Theorem 102 of [Suppes] p. 52.
 
Theoremopabbi2dv 4134* Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2039.
   &       =>   
 
Theoremrelop 4135* A necessary and sufficient condition for a Kuratowski ordered pair to be a relation.
   &       =>   
 
Theoremideqg 4136 For sets, the identity relation is the same as equality. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
 
Theoremideq 4137 For sets, the identity relation is the same as equality.
   =>   
 
Theoremididg 4138 A set is identical to itself. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
 
Theoremopelxpex2 4139 The second member of an ordered pair of classes in a cross product exists when the order pair doesn't belong to . (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
 
Theoremissetid 4140 Two ways of expressing set existence. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
 
Theoremcoss1 4141 Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)
 
Theoremcoss2 4142 Subclass theorem for composition.
 
Theoremcoeq1 4143 Equality theorem for composition of two classes.
 
Theoremcoeq2 4144 Equality theorem for composition of two classes.
 
Theoremcoeq1i 4145 Equality inference for composition of two classes.
   =>   
 
Theoremcoeq2i 4146 Equality inference for composition of two classes.
   =>   
 
Theoremcoeq1d 4147 Equality deduction for composition of two classes.
   =>   
 
Theoremcoeq2d 4148 Equality deduction for composition of two classes.
   =>   
 
Theoremcoeq12i 4149 Equality inference for composition of two classes. (Contributed by FL, 7-Jun-2012.)
   &       =>   
 
Theoremcoeq12d 4150 Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012.)
   &       =>   
 
Theoremhbco 4151* Bound-variable hypothesis builder for function value.
   &       =>   
 
Theoremopelco 4152* Ordered pair membership in a composition. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
   &       =>   
 
Theorembrco 4153* Binary relation on a composition.
   &       =>   
 
Theoremopelco2g 4154* Ordered pair membership in a composition.
 
Theoremcnvss 4155 Subset theorem for converse.
 
Theoremcnveq 4156 Equality theorem for converse.
 
Theoremcnveqi 4157 Equality inference for converse.
   =>   
 
Theoremcnveqd 4158 Equality deduction for converse.
   =>   
 
Theoremelcnv 4159* Membership in a converse. Equation 5 of [Suppes] p. 62.
 
Theoremelcnv2 4160* Membership in a converse. Equation 5 of [Suppes] p. 62.
 
Theoremhbcnv 4161* Bound-variable hypothesis builder for converse.
   =>   
 
Theoremopelcnvg 4162 Ordered-pair membership in converse. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
 
Theorembrcnvg 4163 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61.
 
Theoremopelcnv 4164 Ordered-pair membership in converse.
   &       =>   
 
Theorembrcnv 4165 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61.
   &       =>   
 
Theoremcnvco 4166 Distributive law of converse over class composition. Theorem 26 of [Suppes] p. 64. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
 
Theoremcnvuni 4167* The converse of a class union is the (indexed) union of the converses of its members.
 
Theoremdfdm3 4168* Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24.
 
Theoremdfrn2 4169* Alternate definition of range. Definition 4 of [Suppes] p. 60.
 
Theoremdfrn3 4170* Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24.
 
Theoremdfdm4 4171 Alternate definition of domain.
 
Theoremdfdmf 4172* Definition of domain, using bound-variable hypotheses instead of distinct variable conditions.
   &       =>   
 
Theoremeldmg 4173* Domain membership. Theorem 4 of [Suppes] p. 59.
 
Theoremeldm2g 4174* Domain membership. Theorem 4 of [Suppes] p. 59.
 
Theoremeldm 4175* Membership in a domain. Theorem 4 of [Suppes] p. 59.
   =>   
 
Theoremeldm2 4176* Membership in a domain. Theorem 4 of [Suppes] p. 59.
   =>   
 
Theoremdmss 4177 Subset theorem for domain.
 
Theoremdmeq 4178 Equality theorem for domain.
 
Theoremdmeqi 4179 Equality inference for domain.
   =>   
 
Theoremdmeqd 4180 Equality deduction for domain.
   =>   
 
Theoremopeldm 4181 Membership of first of an ordered pair in a domain.
   =>   
 
Theorembreldm 4182 Membership of first of a binary relation in a domain.
   =>   
 
Theorembreldmg 4183 Membership of first of a binary relation in a domain.
 
Theoremdmun 4184 The domain of a union is the union of domains. Exercise 56(a) of [Enderton] p. 65. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
 
Theoremdmin 4185 The domain of an intersection belong to the intersection of domains. Theorem 6 of [Suppes] p. 60.
 
Theoremdmuni 4186* The domain of a union. Part of Exercise 8 of [Enderton] p. 41.
 
Theoremdmopab 4187* The domain of a class of ordered pairs.
 
Theoremdmopabss 4188* Upper bound for the domain of a restricted class of ordered pairs.
 
Theoremdmopab3 4189* The domain of a restricted class of ordered pairs.
 
Theoremdm0 4190 The domain of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
 
Theoremdmi 4191 The domain of the identity relation is the universe. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
 
Theoremdmv 4192 The domain of the universe is the universe.
 
Theoremdm0rn0 4193 An empty domain implies an empty range.
 
Theoremreldm0 4194 A relation is empty iff its domain is empty.
 
Theoremdmxp 4195 The domain of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
 
Theoremdmxpid 4196 The domain of a square cross product.
 
Theoremdmxpin 4197 The domain of the intersection of two square cross products. Unlike dmin 4185, equality holds.
 
Theoremxpid11 4198 The cross product of a class with itself is one-to-one. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
 
Theoremdmcnvcnv 4199 The domain of the double converse of a class (which doesn't have to be a relation as in dfrel2 4372).
 
Theoremrncnvcnv 4200 The range of the double converse of a class.
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