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Statement List for Metamath Proof Explorer - 4101-4200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxpss12 4101 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 4102 A cross product is included in the ordered pair universe. Exercise 3 of [TakeutiZaring] p. 25.
|- (A X. B) C_ (_V X. _V)
 
Theoremrelxp 4103 A cross product is a relation. Theorem 3.13(i) of [Monk1] p. 37.
|- Rel (A X. B)
 
Theoremxpss1 4104 Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)
|- (A C_ B -> (A X. C) C_ (B X. C))
 
Theoremxpss2 4105 Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)
|- (A C_ B -> (C X. A) C_ (C X. B))
 
Theoremxpsspw 4106 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 4107 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 4108 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 4109 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 4110 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 4111 The union of two relations is a relation. Compare Exercise 5 of [TakeutiZaring] p. 25.
|- (Rel (A u. B) <-> (Rel A /\ Rel B))
 
Theoremrelin1 4112 The intersection with a relation is a relation.
|- (Rel A -> Rel (A i^i B))
 
Theoremrelin2 4113 The intersection with a relation is a relation.
|- (Rel B -> Rel (A i^i B))
 
Theoremreldif 4114 A difference cutting down a relation is a relation.
|- (Rel A -> Rel (A \ B))
 
Theoremreliun 4115 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 4116 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 4117* 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 4118* 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 4119 The empty set is a relation.
|- Rel (/)
 
Theoremrelopabi 4120 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 4121 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 4122* Obsolete proof of relopab 4121 as of 21-Dec-2013.
|- Rel {<.x, y>. | ph}
 
Theoremreli 4123 The identity relation is a relation. Part of Exercise 4.12(p) of [Mendelson] p. 235.
|- Rel _I
 
Theoremrele 4124 The membership relation is a relation.
|- Rel _E
 
Theoremopabid2 4125* A relation expressed as an ordered pair abstraction.
|- (Rel A -> {<.x, y>. | <.x, y>. e. A} = A)
 
Theoreminopab 4126* Intersection of two ordered pair class abstractions.
|- ({<.x, y>. | ph} i^i {<.x, y>. | ps}) = {<.x, y>. | (ph /\ ps)}
 
Theoreminxp 4127 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 4128 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 4129 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 4130* Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2042.
|- Rel A   &   |- (ph -> (<.x, y>. e. A <-> ps))   =>   |- (ph -> A = {<.x, y>. | ps})
 
Theoremrelop 4131* 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 4132 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 4133 For sets, the identity relation is the same as equality.
|- B e. _V   =>   |- (A _I B <-> A = B)
 
Theoremididg 4134 A set is identical to itself. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
|- (A e. V -> A _I A)
 
Theoremopelxpex2 4135 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 4136 Two ways of expressing set existence. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
|- (A e. _V <-> A _I _V)
 
Theoremcoss1 4137 Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)
|- (A C_ B -> (A o. C) C_ (B o. C))
 
Theoremcoss2 4138 Subclass theorem for composition.
|- (A C_ B -> (C o. A) C_ (C o. B))
 
Theoremcoeq1 4139 Equality theorem for composition of two classes.
|- (A = B -> (A o. C) = (B o. C))
 
Theoremcoeq2 4140 Equality theorem for composition of two classes.
|- (A = B -> (C o. A) = (C o. B))
 
Theoremcoeq1i 4141 Equality inference for composition of two classes.
|- A = B   =>   |- (A o. C) = (B o. C)
 
Theoremcoeq2i 4142 Equality inference for composition of two classes.
|- A = B   =>   |- (C o. A) = (C o. B)
 
Theoremcoeq1d 4143 Equality deduction for composition of two classes.
|- (ph -> A = B)   =>   |- (ph -> (A o. C) = (B o. C))
 
Theoremcoeq2d 4144 Equality deduction for composition of two classes.
|- (ph -> A = B)   =>   |- (ph -> (C o. A) = (C o. B))
 
Theoremcoeq12i 4145 Equality inference for composition of two classes. (Contributed by FL, 7-Jun-2012.)
|- A = B   &   |- C = D   =>   |- (A o. C) = (B o. D)
 
Theoremcoeq12d 4146 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 4147* 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 4148* 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 4149* Binary relation on a composition.
|- A e. _V   &   |- B e. _V   =>   |- (A(C o. D)B <-> E.x(ADx /\ xCB))
 
Theoremopelco2g 4150* 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 4151 Subset theorem for converse.
|- (A C_ B -> `'A C_ `'B)
 
Theoremcnveq 4152 Equality theorem for converse.
|- (A = B -> `'A = `'B)
 
Theoremcnveqi 4153 Equality inference for converse.
|- A = B   =>   |- `'A = `'B
 
Theoremcnveqd 4154 Equality deduction for converse.
|- (ph -> A = B)   =>   |- (ph -> `'A = `'B)
 
Theoremelcnv 4155* Membership in a converse. Equation 5 of [Suppes] p. 62.
|- (A e. `'R <-> E.xE.y(A = <.x, y>. /\ yRx))
 
Theoremelcnv2 4156* Membership in a converse. Equation 5 of [Suppes] p. 62.
|- (A e. `'R <-> E.xE.y(A = <.x, y>. /\ <.y, x>. e. R))
 
Theoremhbcnv 4157* Bound-variable hypothesis builder for converse.
|- (y e. A -> A.x y e. A)   =>   |- (y e. `'A -> A.x y e. `'A)
 
Theoremopelcnvg 4158 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 4159 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61.
|- ((A e. C /\ B e. D) -> (A`'RB <-> BRA))
 
Theoremopelcnv 4160 Ordered-pair membership in converse.
|- A e. _V   &   |- B e. _V   =>   |- (<.A, B>. e. `'R <-> <.B, A>. e. R)
 
Theorembrcnv 4161 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61.
|- A e. _V   &   |- B e. _V   =>   |- (A`'RB <-> BRA)
 
Theoremcnvco 4162 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 4163* The converse of a class union is the (indexed) union of the converses of its members.
|- `'U.A = U_x e. A `'x
 
Theoremdfdm3 4164* Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24.
|- dom A = {x | E.y<.x, y>. e. A}
 
Theoremdfrn2 4165* Alternate definition of range. Definition 4 of [Suppes] p. 60.
|- ran A = {y | E.x xAy}
 
Theoremdfrn3 4166* Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24.
|- ran A = {y | E.x<.x, y>. e. A}
 
Theoremdfdm4 4167 Alternate definition of domain.
|- dom A = ran `' A
 
Theoremdfdmf 4168* 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 4169* Domain membership. Theorem 4 of [Suppes] p. 59.
|- (A e. V -> (A e. dom B <-> E.y ABy))
 
Theoremeldm2g 4170* Domain membership. Theorem 4 of [Suppes] p. 59.
|- (A e. V -> (A e. dom B <-> E.y<.A, y>. e. B))
 
Theoremeldm 4171* Membership in a domain. Theorem 4 of [Suppes] p. 59.
|- A e. _V   =>   |- (A e. dom B <-> E.y ABy)
 
Theoremeldm2 4172* Membership in a domain. Theorem 4 of [Suppes] p. 59.
|- A e. _V   =>   |- (A e. dom B <-> E.y<.A, y>. e. B)
 
Theoremdmss 4173 Subset theorem for domain.
|- (A C_ B -> dom A C_ dom B)
 
Theoremdmeq 4174 Equality theorem for domain.
|- (A = B -> dom A = dom B)
 
Theoremdmeqi 4175 Equality inference for domain.
|- A = B   =>   |- dom A = dom B
 
Theoremdmeqd 4176 Equality deduction for domain.
|- (ph -> A = B)   =>   |- (ph -> dom A = dom B)
 
Theoremopeldm 4177 Membership of first of an ordered pair in a domain.
|- A e. _V   =>   |- (<.A, B>. e. C -> A e. dom C)
 
Theorembreldm 4178 Membership of first of a binary relation in a domain.
|- A e. _V   =>   |- (ARB -> A e. dom R)
 
Theorembreldmg 4179 Membership of first of a binary relation in a domain.
|- ((A e. C /\ ARB) -> A e. dom R)
 
Theoremdmun 4180 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 4181 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 4182* The domain of a union. Part of Exercise 8 of [Enderton] p. 41.
|- dom U. A = U_x e. A dom x
 
Theoremdmopab 4183* The domain of a class of ordered pairs.
|- dom {<.x, y>. | ph} = {x | E.yph}
 
Theoremdmopabss 4184* Upper bound for the domain of a restricted class of ordered pairs.
|- dom {<.x, y>. | (x e. A /\ ph)} C_ A
 
Theoremdmopab3 4185* The domain of a restricted class of ordered pairs.
|- (A.x e. A E.yph <-> dom {<.x, y>. | (x e. A /\ ph)} = A)
 
Theoremdm0 4186 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.)
|- dom (/) = (/)
 
Theoremdmi 4187 The domain of the identity relation is the universe. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
|- dom _I = _V
 
Theoremdmv 4188 The domain of the universe is the universe.
|- dom _V = _V
 
Theoremdm0rn0 4189 An empty domain implies an empty range.
|- (dom A = (/) <-> ran A = (/))
 
Theoremreldm0 4190 A relation is empty iff its domain is empty.
|- (Rel A -> (A = (/) <-> dom A = (/)))
 
Theoremdmxp 4191 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.)
|- (B =/= (/) -> dom ( A X. B) = A)
 
Theoremdmxpid 4192 The domain of a square cross product.
|- dom ( A X. A) = A
 
Theoremdmxpin 4193 The domain of the intersection of two square cross products. Unlike dmin 4181, equality holds.
|- dom ((A X. A) i^i (B X. B)) = (A i^i B)
 
Theoremxpid11 4194 The cross product of a class with itself is one-to-one. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
|- ((A X. A) = (B X. B) <-> A = B)
 
Theoremdmcnvcnv 4195 The domain of the double converse of a class (which doesn't have to be a relation as in dfrel2 4364).
|- dom `'`'A = dom A
 
Theoremrncnvcnv 4196 The range of the double converse of a class.
|- ran `'`'A = ran A
 
Theoremelreldm 4197 The first member of an ordered pair in a relation belongs to the domain of the relation.
|- ((Rel A /\ B e. A) -> |^||^|B e. dom A)
 
Theoremrneq 4198 Equality theorem for range.
|- (A = B -> ran A = ran B)
 
Theoremrneqi 4199 Equality inference for range.
|- A = B   =>   |- ran A = ran B
 
Theoremrneqd 4200 Equality deduction for range.
|- (ph -> A = B)   =>   |- (ph -> ran A = ran B)
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