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Statement List for Metamath Proof Explorer - 4101-4200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrelss 4101 Subclass theorem for relation predicate. Theorem 2 of [Suppes] p. 58.
|- (A C_ B -> (Rel B -> Rel A))
 
Theoremssrel 4102* A subclass relationship depends only on a relation's ordered pairs. Theorem 3.2(i) of [Monk1] p. 33. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
|- (Rel A -> (A C_ B <-> A.xA.y(<.x, y>. e. A -> <.x, y>. e. B)))
 
Theoremeqrel 4103* Extensionality principle for relations. Theorem 3.2(ii) of [Monk1] p. 33.
|- ((Rel A /\ Rel B) -> (A = B <-> A.xA.y(<.x, y>. e. A <-> <.x, y>. e. B)))
 
Theoremrelssi 4104* Inference from subclass principle for relations.
|- Rel A   &   |- (<.x, y>. e. A -> <.x, y>. e. B)   =>   |- A C_ B
 
Theoremrelssdv 4105* Deduction from subclass principle for relations.
|- (ph -> Rel A)   &   |- (ph -> (<.x, y>. e. A -> <.x, y>. e. B))   =>   |- (ph -> A C_ B)
 
Theoremeqrelriv 4106* Inference from extensionality principle for relations. (Contributed by FL, 15-Oct-2012.)
|- (<.x, y>. e. A <-> <.x, y>. e. B)   =>   |- ((Rel A /\ Rel B) -> A = B)
 
Theoremeqrelriiv 4107* Inference from extensionality principle for relations.
|- Rel A   &   |- Rel B   &   |- (<.x, y>. e. A <-> <.x, y>. e. B)   =>   |- A = B
 
Theoremeqbrriv 4108* Inference from extensionality principle for relations.
|- Rel A   &   |- Rel B   &   |- (xAy <-> xBy)   =>   |- A = B
 
Theoremeqrelrdv 4109* Deduce equality of relations from equivalence of membership. (Contributed by Rodolfo Medina, 10-Oct-2010.)
|- Rel A   &   |- Rel B   &   |- (ph -> (<.x, y>. e. A <-> <.x, y>. e. B))   =>   |- (ph -> A = B)
 
Theoremeqrelrdv2 4110* A version of eqrelrdv 4109. (Contributed by Rodolfo Medina, 10-Oct-2010.)
|- (ph -> (<.x, y>. e. A <-> <.x, y>. e. B))   =>   |- (((Rel A /\ Rel B) /\ ph) -> A = B)
 
Theoremssrelrel 4111* A subclass relationship determined by ordered triples. Use relrelss 4434 to express the antecedent in terms of the relation predicate. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
|- (A C_ ((_V X. _V) X. _V) -> (A C_ B <-> A.xA.yA.z(<.<.x, y>., z>. e. A -> <.<.x, y>., z>. e. B)))
 
Theoremeqrelrel 4112* Extensionality principle for ordered triples (used by 2-place operations df-oprab 4931), analogous to eqrel 4103. Use relrelss 4434 to express the antecedent in terms of the relation predicate.
|- ((A u. B) C_ ((_V X. _V) X. _V) -> (A = B <-> A.xA.yA.z(<.<.x, y>., z>. e. A <-> <.<.x, y>., z>. e. B)))
 
Theoremelrel 4113* A member of a relation is an ordered pair.
|- ((Rel R /\ A e. R) -> E.xE.y A = <.x, y>.)
 
Theoremrelsn 4114 A singleton is a relation iff it is an ordered pair.
|- A e. _V   =>   |- (Rel {A} <-> A e. (_V X. _V))
 
Theoremrelsnop 4115 A singleton of an ordered pair is a relation.
|- A e. _V   =>   |- Rel {<.A, B>.}
 
Theoremxpss12 4116 Subset theorem for cross product. Generalization of Theorem 101 of [Suppes] p. 52. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
|- ((A C_ B /\ C C_ D) -> (A X. C) C_ (B X. D))
 
Theoremxpss 4117 A cross product is included in the ordered pair universe. Exercise 3 of [TakeutiZaring] p. 25.
|- (A X. B) C_ (_V X. _V)
 
Theoremrelxp 4118 A cross product is a relation. Theorem 3.13(i) of [Monk1] p. 37.
|- Rel (A X. B)
 
Theoremxpss1 4119 Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)
|- (A C_ B -> (A X. C) C_ (B X. C))
 
Theoremxpss2 4120 Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)
|- (A C_ B -> (C X. A) C_ (C X. B))
 
Theoremxpsspw 4121 A cross product is included in the power of the power of the union of its arguments.
|- (A X. B) C_ ~P~P(A u. B)
 
TheoremxpsspwOLD 4122 A cross product is included in the power of the power of the union of its arguments.
|- (A X. B) C_ ~P~P(A u. B)
 
Theoremunixpss 4123 The double class union of a cross product is included in the union of its arguments.
|- U.U.(A X. B) C_ (A u. B)
 
Theoremxpexg 4124 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23.
|- ((A e. V /\ B e. W) -> (A X. B) e. _V)
 
Theoremxpex 4125 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23.
|- A e. _V   &   |- B e. _V   =>   |- (A X. B) e. _V
 
Theoremrelun 4126 The union of two relations is a relation. Compare Exercise 5 of [TakeutiZaring] p. 25.
|- (Rel (A u. B) <-> (Rel A /\ Rel B))
 
Theoremrelin1 4127 The intersection with a relation is a relation.
|- (Rel A -> Rel (A i^i B))
 
Theoremrelin2 4128 The intersection with a relation is a relation.
|- (Rel B -> Rel (A i^i B))
 
Theoremreldif 4129 A difference cutting down a relation is a relation.
|- (Rel A -> Rel (A \ B))
 
Theoremreliun 4130 An indexed union is a relation iff each member of its indexed family is a relation.
|- (Rel U_x e. A B <-> A.x e. A Rel B)
 
Theoremreliin 4131 An indexed intersection is a relation if if at least one of the member of the indexed family is a relation.
|- (E.x e. A Rel B -> Rel |^|_x e. A B)
 
Theoremreluni 4132* The union of a class is a relation iff any member is a relation. Exercise 6 of [TakeutiZaring] p. 25 and its converse.
|- (Rel U.A <-> A.x e. A Rel x)
 
Theoremrelint 4133* The intersection of a class is a relation if at least one member is a relation.
|- (E.x e. A Rel x -> Rel |^|A)
 
Theoremrel0 4134 The empty set is a relation.
|- Rel (/)
 
Theoremrelopabi 4135 A class of ordered pairs is a relation. (The proof was shortened by Mario Carneiro, 21-Dec-2013.)
|- A = {<.x, y>. | ph}   =>   |- Rel A
 
Theoremrelopab 4136 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.)
|- Rel {<.x, y>. | ph}
 
TheoremrelopabOLD 4137* Obsolete proof of relopab 4136 as of 21-Dec-2013.
|- Rel {<.x, y>. | ph}
 
Theoremreli 4138 The identity relation is a relation. Part of Exercise 4.12(p) of [Mendelson] p. 235.
|- Rel _I
 
Theoremrele 4139 The membership relation is a relation.
|- Rel _E
 
Theoremopabid2 4140* A relation expressed as an ordered pair abstraction.
|- (Rel A -> {<.x, y>. | <.x, y>. e. A} = A)
 
Theoreminopab 4141* Intersection of two ordered pair class abstractions.
|- ({<.x, y>. | ph} i^i {<.x, y>. | ps}) = {<.x, y>. | (ph /\ ps)}
 
Theoreminxp 4142 The intersection of two cross products. Exercise 9 of [TakeutiZaring] p. 25. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
|- ((A X. B) i^i (C X. D)) = ((A i^i C) X. (B i^i D))
 
Theoremxpindi 4143 Distributive law for cross product over intersection. Theorem 102 of [Suppes] p. 52.
|- (A X. (B i^i C)) = ((A X. B) i^i (A X. C))
 
Theoremxpindir 4144 Distributive law for cross product over intersection. Similar to Theorem 102 of [Suppes] p. 52.
|- ((A i^i B) X. C) = ((A X. C) i^i (B X. C))
 
Theoremopabbi2dv 4145* Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2070.
|- Rel A   &   |- (ph -> (<.x, y>. e. A <-> ps))   =>   |- (ph -> A = {<.x, y>. | ps})
 
Theoremrelop 4146* A necessary and sufficient condition for a Kuratowski ordered pair to be a relation.
|- A e. _V   &   |- B e. _V   =>   |- (Rel <.A, B>. <-> E.xE.y(A = {x} /\ B = {x, y}))
 
Theoremideqg 4147 For sets, the identity relation is the same as equality. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
|- (B e. V -> (A _I B <-> A = B))
 
Theoremideq 4148 For sets, the identity relation is the same as equality.
|- B e. _V   =>   |- (A _I B <-> A = B)
 
Theoremididg 4149 A set is identical to itself. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
|- (A e. V -> A _I A)
 
Theoremopelxpex2 4150 The second member of an ordered pair of classes in a cross product exists when the order pair doesn't belong to _I. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
|- (<.A, B>. e. ((C X. D) \ _I ) -> B e. _V)
 
Theoremissetid 4151 Two ways of expressing set existence. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
|- (A e. _V <-> A _I _V)
 
Theoremcoss1 4152 Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)
|- (A C_ B -> (A o. C) C_ (B o. C))
 
Theoremcoss2 4153 Subclass theorem for composition.
|- (A C_ B -> (C o. A) C_ (C o. B))
 
Theoremcoeq1 4154 Equality theorem for composition of two classes.
|- (A = B -> (A o. C) = (B o. C))
 
Theoremcoeq2 4155 Equality theorem for composition of two classes.
|- (A = B -> (C o. A) = (C o. B))
 
Theoremcoeq1i 4156 Equality inference for composition of two classes.
|- A = B   =>   |- (A o. C) = (B o. C)
 
Theoremcoeq2i 4157 Equality inference for composition of two classes.
|- A = B   =>   |- (C o. A) = (C o. B)
 
Theoremcoeq1d 4158 Equality deduction for composition of two classes.
|- (ph -> A = B)   =>   |- (ph -> (A o. C) = (B o. C))
 
Theoremcoeq2d 4159 Equality deduction for composition of two classes.
|- (ph -> A = B)   =>   |- (ph -> (C o. A) = (C o. B))
 
Theoremcoeq12i 4160 Equality inference for composition of two classes. (Contributed by FL, 7-Jun-2012.)
|- A = B   &   |- C = D   =>   |- (A o. C) = (B o. D)
 
Theoremcoeq12d 4161 Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012.)
|- (ph -> A = B)   &   |- (ph -> C = D)   =>   |- (ph -> (A o. C) = (B o. D))
 
Theoremhbco 4162* Bound-variable hypothesis builder for function value.
|- (y e. A -> A.x y e. A)   &   |- (y e. B -> A.x y e. B)   =>   |- (y e. (A o. B) -> A.x y e. (A o. B))
 
Theoremopelco 4163* Ordered pair membership in a composition. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
|- A e. _V   &   |- B e. _V   =>   |- (<.A, B>. e. (C o. D) <-> E.x(ADx /\ xCB))
 
Theorembrco 4164* Binary relation on a composition.
|- A e. _V   &   |- B e. _V   =>   |- (A(C o. D)B <-> E.x(ADx /\ xCB))
 
Theoremopelco2g 4165* Ordered pair membership in a composition.
|- ((A e. V /\ B e. W) -> (<.A, B>. e. (C o. D) <-> E.x(<.A, x>. e. D /\ <.x, B>. e. C)))
 
Theoremcnvss 4166 Subset theorem for converse.
|- (A C_ B -> `'A C_ `'B)
 
Theoremcnveq 4167 Equality theorem for converse.
|- (A = B -> `'A = `'B)
 
Theoremcnveqi 4168 Equality inference for converse.
|- A = B   =>   |- `'A = `'B
 
Theoremcnveqd 4169 Equality deduction for converse.
|- (ph -> A = B)   =>   |- (ph -> `'A = `'B)
 
Theoremelcnv 4170* Membership in a converse. Equation 5 of [Suppes] p. 62.
|- (A e. `'R <-> E.xE.y(A = <.x, y>. /\ yRx))
 
Theoremelcnv2 4171* Membership in a converse. Equation 5 of [Suppes] p. 62.
|- (A e. `'R <-> E.xE.y(A = <.x, y>. /\ <.y, x>. e. R))
 
Theoremhbcnv 4172* Bound-variable hypothesis builder for converse.
|- (y e. A -> A.x y e. A)   =>   |- (y e. `'A -> A.x y e. `'A)
 
Theoremopelcnvg 4173 Ordered-pair membership in converse. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
|- ((A e. C /\ B e. D) -> (<.A, B>. e. `'R <-> <.B, A>. e. R))
 
Theorembrcnvg 4174 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61.
|- ((A e. C /\ B e. D) -> (A`'RB <-> BRA))
 
Theoremopelcnv 4175 Ordered-pair membership in converse.
|- A e. _V   &   |- B e. _V   =>   |- (<.A, B>. e. `'R <-> <.B, A>. e. R)
 
Theorembrcnv 4176 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61.
|- A e. _V   &   |- B e. _V   =>   |- (A`'RB <-> BRA)
 
Theoremcnvco 4177 Distributive law of converse over class composition. Theorem 26 of [Suppes] p. 64. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
|- `'(A o. B) = (`'B o. `'A)
 
Theoremcnvuni 4178* The converse of a class union is the (indexed) union of the converses of its members.
|- `'U.A = U_x e. A `'x
 
Theoremdfdm3 4179* Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24.
|- dom A = {x | E.y<.x, y>. e. A}
 
Theoremdfrn2 4180* Alternate definition of range. Definition 4 of [Suppes] p. 60.
|- ran A = {y | E.x xAy}
 
Theoremdfrn3 4181* Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24.
|- ran A = {y | E.x<.x, y>. e. A}
 
Theoremdfdm4 4182 Alternate definition of domain.
|- dom A = ran `' A
 
Theoremdfdmf 4183* Definition of domain, using bound-variable hypotheses instead of distinct variable conditions.
|- (z e. A -> A.x z e. A)   &   |- (z e. A -> A.y z e. A)   =>   |- dom A = {x | E.y<.x, y>. e. A}
 
Theoremeldmg 4184* Domain membership. Theorem 4 of [Suppes] p. 59.
|- (A e. V -> (A e. dom B <-> E.y ABy))
 
Theoremeldm2g 4185* Domain membership. Theorem 4 of [Suppes] p. 59.
|- (A e. V -> (A e. dom B <-> E.y<.A, y>. e. B))
 
Theoremeldm 4186* Membership in a domain. Theorem 4 of [Suppes] p. 59.
|- A e. _V   =>   |- (A e. dom B <-> E.y ABy)
 
Theoremeldm2 4187* Membership in a domain. Theorem 4 of [Suppes] p. 59.
|- A e. _V   =>   |- (A e. dom B <-> E.y<.A, y>. e. B)
 
Theoremdmss 4188 Subset theorem for domain.
|- (A C_ B -> dom A C_ dom B)
 
Theoremdmeq 4189 Equality theorem for domain.
|- (A = B -> dom A = dom B)
 
Theoremdmeqi 4190 Equality inference for domain.
|- A = B   =>   |- dom A = dom B
 
Theoremdmeqd 4191 Equality deduction for domain.
|- (ph -> A = B)   =>   |- (ph -> dom A = dom B)
 
Theoremopeldm 4192 Membership of first of an ordered pair in a domain.
|- A e. _V   =>   |- (<.A, B>. e. C -> A e. dom C)
 
Theorembreldm 4193 Membership of first of a binary relation in a domain.
|- A e. _V   =>   |- (ARB -> A e. dom R)
 
Theorembreldmg 4194 Membership of first of a binary relation in a domain.
|- ((A e. C /\ ARB) -> A e. dom R)
 
Theoremdmun 4195 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.)
|- dom ( A u. B) = (dom A u. dom B)
 
Theoremdmin 4196 The domain of an intersection belong to the intersection of domains. Theorem 6 of [Suppes] p. 60.
|- dom ( A i^i B) C_ (dom A i^i dom B)
 
Theoremdmuni 4197* The domain of a union. Part of Exercise 8 of [Enderton] p. 41.
|- dom U. A = U_x e. A dom x
 
Theoremdmopab 4198* The domain of a class of ordered pairs.
|- dom {<.x, y>. | ph} = {x | E.yph}
 
Theoremdmopabss 4199* Upper bound for the domain of a restricted class of ordered pairs.
|- dom {<.x, y>. | (x e. A /\ ph)} C_ A
 
Theoremdmopab3 4200* The domain of a restricted class of ordered pairs.
|- (A.x e. A E.yph <-> dom {<.x, y>. | (x e. A /\ ph)} = A)
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