mmtheorems36 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 3501-3600 - Page 36 of 123

Type	Label	Description
Statement
Theorem	elima2 3501	Membership in an image. Theorem 34 of [Suppes] p. 65.
|- A e. V   =>   |- (A e. (B"C) <-> E.x(x e. C /\ xBA))
Theorem	elima3 3502	Membership in an image. Theorem 34 of [Suppes] p. 65.
|- A e. V   =>   |- (A e. (B"C) <-> E.x(x e. C /\ <.x, A>. e. B))
Theorem	hbima 3503	Bound-variable hypothesis builder for image.
|- (y e. A -> A.x y e. A)   &   |- (y e. B -> A.x y e. B)   =>   |- (y e. (A"B) -> A.x y e. (A"B))
Theorem	hbimad 3504	Deduction version of bound-variable hypothesis builder hbima 3503. (Contributed by FL, 15-Dec-2006.)
|- (ph -> A.xph)   &   |- (ph -> (y e. A -> A.x y e. A))   &   |- (ph -> (y e. B -> A.x y e. B))   =>   |- (ph -> (y e. (A"B) -> A.x y e. (A"B)))
Theorem	csbima12g 3505	Move class substitution in and out of the image of a function. (Contributed by FL, 15-Dec-2006.)
|- (A e. C -> [_A / x]_(F"B) = ([_A / x]_F"[_A / x]_B))
Theorem	imadmrn 3506	The image of the domain of a class is the range of the class.
|- (A"dom A) = ran A
Theorem	imassrn 3507	The image of a class is a subset of its range. Theorem 3.16(xi) of [Monk1] p. 39.
|- (A"B) (_ ran A
Theorem	imaexg 3508	The image of a set is a set. Theorem 3.17 of [Monk1] p. 39.
|- (A e. C -> (A"B) e. V)
Theorem	imai 3509	Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38.
|- (I"A) = A
Theorem	rnresi 3510	The range of the restricted identity function.
|- ran ( I |` A) = A
Theorem	resiima 3511	The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)
|- (B (_ A -> ((I |` A)"B) = B)
Theorem	ima0 3512	Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38.
|- (A"(/)) = (/)
Theorem	0ima 3513	Image under the empty relation. (Contributed by FL, 11-Jan-2007.)
|- ((/)"A) = (/)
Theorem	imadisj 3514	A class whose image under another is empty is disjoint with the other's domain. (Contributed by FL, 24-Jan-2007.)
|- ((A"B) = (/) <-> (dom A i^i B) = (/))
Theorem	cnvimass 3515	A pre-image under any class is included in the domain of the class. (Contributed by FL, 29-Jan-2007.)
|- (`'A"B) (_ dom A
Theorem	imasng 3516	The image of a singleton.
|- (A e. B -> (R"{A}) = {y | ARy})
Theorem	relimasn 3517	The image of a singleton.
|- (Rel R -> (R"{A}) = {y | ARy})
Theorem	elimasn 3518	Membership in an image of a singleton.
|- B e. V   &   |- C e. V   =>   |- (C e. (A"{B}) <-> <.B, C>. e. A)
Theorem	elimasng 3519	Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.)
|- ((B e. R /\ C e. S) -> (C e. (A"{B}) <-> <.B, C>. e. A))
Theorem	args 3520	Two ways to express the class of unique-valued arguments of F, which is the same as the domain of F whenever F is a function. The left-hand side of the equality is from Definition 10.2 of [Quine] p. 65. Quine uses the notation "arg F" for this class (for which we have no separate notation). Observe the resemblance to our df-fv 3279, which was based on the idea in Quine's definition.
|- {x | E.y(F"{x}) = {y}} = {x | E!y xFy}
Theorem	eliniseg 3521	Membership in an initial segment. The idiom (`'A"{B}), meaning {x | xAB}, is used to specify an initial segment in (for example) Definition 6.21 of [TakeutiZaring] p. 30.
|- C e. V   =>   |- (B e. D -> (C e. (`'A"{B}) <-> CAB))
Theorem	iniseg 3522	An idiom that signifies an initial segment of an ordering, used, for example, in Definition 6.21 of [TakeutiZaring] p. 30.
|- (B e. C -> (`'A"{B}) = {x | xAB})
Theorem	dffr3 3523	Alternate definition of founded relation. Definition 6.21 of [TakeutiZaring] p. 30.
|- (R Fr A <-> A.x((x (_ A /\ x =/= (/)) -> E.y e. x (x i^i (`'R"{y})) = (/)))
Theorem	imass1 3524	Subset theorem for image.
|- (A (_ B -> (A"C) (_ (B"C))
Theorem	imass2 3525	Subset theorem for image. Exercise 22(a) of [Enderton] p. 53.
|- (A (_ B -> (C"A) (_ (C"B))
Theorem	ndmima 3526	The image of a singleton outside the domain is empty.
|- (-. A e. dom B -> (B"{A}) = (/))
Theorem	relcnv 3527	A converse is a relation. Theorem 12 of [Suppes] p. 62.
|- Rel `'A
Theorem	cotr 3528	Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51.
|- ((R o. R) (_ R <-> A.xA.yA.z((xRy /\ yRz) -> xRz))
Theorem	cnvsym 3529	Two ways of saying a relation is symmetric. Similar to definition of symmetry in [Schechter] p. 51.
|- (`'R (_ R <-> A.xA.y(xRy -> yRx))
Theorem	intasym 3530	Two ways of saying a relation is antisymmetric. Definition of antisymmetry in [Schechter] p. 51.
|- ((R i^i `'R) (_ I <-> A.xA.y((xRy /\ yRx) -> x = y))
Theorem	asymref 3531	Two ways of saying a relation is antisymmetric and reflexive. U.U.R is the field of a relation by relfld 3620.
|- ((R i^i `'R) = (I |` U.U.R) <-> A.x e. U.U.RA.y((xRy /\ yRx) <-> x = y))
Theorem	asymref2 3532	Two ways of saying a relation is antisymmetric and reflexive.
|- ((R i^i `'R) = (I |` U.U.R) <-> (A.x e. U.U.RxRx /\ A.xA.y((xRy /\ yRx) -> x = y)))
Theorem	intirr 3533	Two ways of saying a relation is irreflexive. Definition of irreflexivity in [Schechter] p. 51.
|- ((R i^i I) = (/) <-> A.x -. xRx)
Theorem	soirri 3534	A strict order relation is irreflexive.
|- A e. V   &   |- R Or S   &   |- R (_ (S X. S)   =>   |- -. ARA
Theorem	sotri 3535	A strict order relation is a transitive relation.
|- A e. V   &   |- R Or S   &   |- R (_ (S X. S)   &   |- B e. V   &   |- C e. V   =>   |- ((ARB /\ BRC) -> ARC)
Theorem	son2lpi 3536	A strict order relation has no 2-cycle loops.
|- A e. V   &   |- R Or S   &   |- R (_ (S X. S)   &   |- B e. V   =>   |- -. (ARB /\ BRA)
Theorem	cnvopab 3537	The converse of a class abstraction of ordered pairs.
|- `'{<.x, y>. | ph} = {<.y, x>. | ph}
Theorem	cnv0 3538	The converse of the empty set.
|- `'(/) = (/)
Theorem	cnvi 3539	The converse of the identity relation. Theorem 3.7(ii) of [Monk1] p. 36.
|- `'I = I
Theorem	cnvun 3540	The converse of a union is the union of converses. Theorem 16 of [Suppes] p. 62.
|- `'(A u. B) = (`'A u. `'B)
Theorem	cnvin 3541	Distributive law for converse over intersection. Theorem 15 of [Suppes] p. 62.
|- `'(A i^i B) = (`'A i^i `'B)
Theorem	rnun 3542	Distributive law for range over union. Theorem 8 of [Suppes] p. 60.
|- ran ( A u. B) = (ran A u. ran B)
Theorem	rnin 3543	The range of an intersection belongs the intersection of ranges. Theorem 9 of [Suppes] p. 60.
|- ran ( A i^i B) (_ (ran A i^i ran B)
Theorem	rnuni 3544	The range of a union. Part of Exercise 8 of [Enderton] p. 41.
|- ran U. A = U_x e. A ran x
Theorem	imaundi 3545	Distributive law for image over union. Theorem 35 of [Suppes] p. 65.
|- (A"(B u. C)) = ((A"B) u. (A"C))
Theorem	imaundir 3546	The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)
|- ((A u. B)"C) = ((A"C) u. (B"C))
Theorem	dminss 3547	An upper bound for intersection with a domain. Theorem 40 of [Suppes] p. 66, who calls it "somewhat surprising."
|- (dom R i^i A) (_ (`'R"(R"A))
Theorem	imainss 3548	An upper bound for intersection with an image. Theorem 41 of [Suppes] p. 66.
|- ((R"A) i^i B) (_ (R"(A i^i (`'R"B)))
Theorem	cnvxp 3549	The converse of a cross product. Exercise 11 of [Suppes] p. 67.
|- `'(A X. B) = (B X. A)
Theorem	xp0 3550	The cross product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37.
|- (A X. (/)) = (/)
Theorem	xpnz 3551	The cross product of nonempty classes is nonempty. (Variation of a theorem contributed by Raph Levien, 30-Jun-2006.)
|- ((A =/= (/) /\ B =/= (/)) <-> (A X. B) =/= (/))
Theorem	xpeq0 3552	At least one member of an empty cross product is empty.
|- ((A X. B) = (/) <-> (A = (/) \/ B = (/)))
Theorem	xpdisj1 3553	Cross products with disjoint sets are disjoint.
|- ((A i^i B) = (/) -> ((A X. C) i^i (B X. D)) = (/))
Theorem	xpdisj2 3554	Cross products with disjoint sets are disjoint.
|- ((A i^i B) = (/) -> ((C X. A) i^i (D X. B)) = (/))
Theorem	xpsndisj 3555	Cross products with two different singletons are disjoint.
|- (B =/= D -> ((A X. {B}) i^i (C X. {D})) = (/))
Theorem	resdisj 3556	A double restriction to disjoint classes is the empty set.
|- ((A i^i B) = (/) -> ((C |` A) |` B) = (/))
Theorem	rnxp 3557	The range of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37.
|- (A =/= (/) -> ran ( A X. B) = B)
Theorem	dmxpss 3558	The domain of a cross product is a subclass of the first factor.
|- dom ( A X. B) (_ A
Theorem	rnxpss 3559	The range of a cross product is a subclass of the second factor.
|- ran ( A X. B) (_ B
Theorem	ssxpb 3560	A cross-product subclass relationship is equivalent to the relationship for it components.
|- ((A X. B) =/= (/) -> ((A X. B) (_ (C X. D) <-> (A (_ C /\ B (_ D)))
Theorem	xp11 3561	The cross product of non-empty classes is one-to-one.
|- ((A =/= (/) /\ B =/= (/)) -> ((A X. B) = (C X. D) <-> (A = C /\ B = D)))
Theorem	xp11a 3562	The first argument of a cross product is one-to-one.
|- (A =/= (/) -> ((A X. A) = (B X. A) <-> A = B))
Theorem	xp11b 3563	The second argument of a cross product is one-to-one.
|- (A =/= (/) -> ((A X. A) = (A X. B) <-> A = B))
Theorem	xpexr 3564	If a cross product is a set, one of its components must be a set.
|- ((A X. B) e. C -> (A e. V \/ B e. V))
Theorem	xpexr2 3565	If a nonempty cross product is a set, so are both of its components.
|- (((A X. B) e. C /\ (A X. B) =/= (/)) -> (A e. V /\ B e. V))
Theorem	ssrnres 3566	Subset of the range of a restriction.
|- (B (_ ran ( C |` A) <-> ran ( C i^i (A X. B)) = B)
Theorem	rninxp 3567	Range of the intersection with a cross product.
|- (ran ( C i^i (A X. B)) = B <-> A.y e. B E.x e. A xCy)
Theorem	dminxp 3568	Domain of the intersection with a cross product.
|- (dom ( C i^i (A X. B)) = A <-> A.x e. A E.y e. B xCy)
Theorem	dfrel2 3569	Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25.
|- (Rel R <-> `'`'R = R)
Theorem	cnvcnv 3570	The double converse of a class strips out all elements that are not ordered pairs.
|- `'`'A = (A i^i (V X. V))
Theorem	cnvcnv2 3571	The double converse of a class equals its restriction to the universe.
|- `'`'A = (A |` V)
Theorem	cnvcnvss 3572	The double converse of a class is a subclass. Exercise 2 of [TakeutiZaring] p. 25.
|- `'`'A (_ A
Theorem	dmsnn0 3573	The domain of a singleton is nonzero iff the singleton argument is an ordered pair.
|- (A e. (V X. V) <-> dom { A} =/= (/))
Theorem	rnsnn0 3574	The range of a singleton is nonzero iff the singleton argument is an ordered pair.
|- (A e. (V X. V) <-> ran { A} =/= (/))
Theorem	dmsn0 3575	The domain of the singleton of the empty set is empty.
|- dom {(/)} = (/)
Theorem	dmsn0el 3576	The domain of a singleton is empty if the singleton's argument contains the empty set.
|- ((/) e. A -> dom { A} = (/))
Theorem	dmsnop 3577	The domain of a singleton of an ordered pair is the singleton of the first member.
|- dom {<.A, B>.} = {A}
Theorem	dmsnsnsn 3578	The domain of the singleton of the singleton of a singleton.
|- dom {{{A}}} = {A}
Theorem	op1sta 3579	Extract the first member of an ordered pair. (See op2nda 3584 to extract the second member, op1stb 3136 for an alternate version, and op1st 4146 for the preferred version..) (Contributed by Raph Levien, 4-Dec-2003.)
|- A e. V   =>   |- U.dom {<.A, B>.} = A
Theorem	cnvsn 3580	Converse of a singleton of an ordered pair.
|- A e. V   &   |- B e. V   =>   |- `'{<.A, B>.} = {<.B, A>.}
Theorem	opswap 3581	Swap the members of an ordered pair.
|- A e. V   &   |- B e. V   =>   |- U.`'{<.A, B>.} = <.B, A>.
Theorem	rnsnop 3582	The range of a singleton of an ordered pair is the singleton of the second member.
|- A e. V   &   |- B e. V   =>   |- ran {<.A, B>.} = {B}
Theorem	op2ndb 3583	Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 3136 to extract the first member, op2nda 3584 for an alternate version, and op2nd 4147 for the preferred version.)
|- A e. V   &   |- B e. V   =>   |- |^||^||^|`'{<.A, B>.} = B
Theorem	op2nda 3584	Extract the second member of an ordered pair. (See op1sta 3579 to extract the first member, op2ndb 3583 for an alternate version, and op2nd 4147 for the preferred version.)
|- A e. V   &   |- B e. V   =>   |- U.ran {<.A, B>.} = B
Theorem	elxp4 3585	Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 3586, elxp6 4161, and elxp7 4162.
|- (A e. (B X. C) <-> (A = <.U.dom { A}, U.ran { A}>. /\ (U.dom { A} e. B /\ U.ran { A} e. C)))
Theorem	elxp5 3586	Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 3585 when the double intersection does not create class existence problems (caused by int0 2614).
|- (A e. (B X. C) <-> (A = <.|^||^|A, U.ran { A}>. /\ (|^||^|A e. B /\ U.ran { A} e. C)))
Theorem	dfrel3 3587	Alternate definition of relation.
|- (Rel R <-> (R |` V) = R)
Theorem	dmresv 3588	The domain of a universal restriction.
|- dom ( A |` V) = dom A
Theorem	rnresv 3589	The range of a universal restriction.
|- ran ( A |` V) = ran A
Theorem	dfrn4 3590	Range defined in terms of image.
|- ran A = (A"V)
Theorem	rescnvcnv 3591	The restriction of the double converse of a class.
|- (`'`'A |` B) = (A |` B)
Theorem	cnvcnvres 3592	The double converse of the restriction of a class.
|- `'`'(A |` B) = (`'`'A |` B)
Theorem	imacnvcnv 3593	The image of the double converse of a class.
|- (`'`'A"B) = (A"B)
Theorem	resdm2 3594	A class restricted to its domain equals its double converse.
|- (A |` dom A) = `'`'A
Theorem	resdmres 3595	Restriction to the domain of a restriction.
|- (A |` dom ( A |` B)) = (A |` B)
Theorem	imadmres 3596	The image of the domain of a restriction.
|- (A"dom ( A |` B)) = (A"B)
Theorem	relco 3597	A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25.
|- Rel (A o. B)
Theorem	dfco2 3598	Alternate definition of a class composition, using only one bound variable.
|- (A o. B) = U_x e. V ((`'B"{x}) X. (A"{x}))
Theorem	dfco2a 3599	Generalization of dfco2 3598, where C can have any value between dom A i^i ran B and V.
|- ((dom A i^i ran B) (_ C -> (A o. B) = U_x e. C ((`'B"{x}) X. (A"{x})))
Theorem	coundi 3600	Class composition distributes over union.
|- (A o. (B u. C)) = ((A o. B) u. (A o. C))