mmtheorems35 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 3401-3500 - Page 35 of 108

Type	Label	Description
Statement
Theorem	imaeq1 3401	Equality theorem for image.
|- (A = B -> (A"C) = (B"C))
Theorem	imaeq2 3402	Equality theorem for image.
|- (A = B -> (C"A) = (C"B))
Theorem	imaeq1d 3403	Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
|- (ph -> A = B)   =>   |- (ph -> (A"C) = (B"C))
Theorem	imaeq2d 3404	Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
|- (ph -> A = B)   =>   |- (ph -> (C"A) = (C"B))
Theorem	dfima2 3405	Alternate definition of image. Compare definition (d) of [Enderton] p. 44.
|- (A"B) = {y | E.x e. B xAy}
Theorem	dfima3 3406	Alternate definition of image. Compare definition (d) of [Enderton] p. 44.
|- (A"B) = {y | E.x(x e. B /\ <.x, y>. e. A)}
Theorem	elimag 3407	Membership in an image. Theorem 34 of [Suppes] p. 65.
|- (A e. D -> (A e. (B"C) <-> E.x e. C xBA))
Theorem	elima 3408	Membership in an image. Theorem 34 of [Suppes] p. 65.
|- A e. V   =>   |- (A e. (B"C) <-> E.x e. C xBA)
Theorem	elima2 3409	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 3410	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 3411	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 3412	Deduction version of bound-variable hypothesis builder hbima 3411. (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 3413	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 3414	The image of the domain of a class is the range of the class.
|- (A"dom A) = ran A
Theorem	imassrn 3415	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 3416	The image of a set is a set. Theorem 3.17 of [Monk1] p. 39.
|- (A e. C -> (A"B) e. V)
Theorem	imai 3417	Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38.
|- (I"A) = A
Theorem	rnresi 3418	The range of the restricted identity function.
|- ran ( I |` A) = A
Theorem	resiima 3419	The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)
|- (B (_ A -> ((I |` A)"B) = B)
Theorem	ima0 3420	Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38.
|- (A"(/)) = (/)
Theorem	0ima 3421	Image under the empty relation. (Contributed by FL, 11-Jan-2007.)
|- ((/)"A) = (/)
Theorem	imadisj 3422	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 3423	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 3424	The image of a singleton.
|- (A e. B -> (R"{A}) = {y | ARy})
Theorem	relimasn 3425	The image of a singleton.
|- (Rel R -> (R"{A}) = {y | ARy})
Theorem	elimasn 3426	Membership in an image of a singleton.
|- B e. V   &   |- C e. V   =>   |- (C e. (A"{B}) <-> <.B, C>. e. A)
Theorem	elimasng 3427	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 3428	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 3198, which was based on the idea in Quine's definition.
|- {x | E.y(F"{x}) = {y}} = {x | E!y xFy}
Theorem	eliniseg 3429	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 3430	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 3431	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 3432	Subset theorem for image.
|- (A (_ B -> (A"C) (_ (B"C))
Theorem	imass2 3433	Subset theorem for image. Exercise 22(a) of [Enderton] p. 53.
|- (A (_ B -> (C"A) (_ (C"B))
Theorem	ndmima 3434	The image of a singleton outside the domain is empty.
|- (-. A e. dom B -> (B"{A}) = (/))
Theorem	relcnv 3435	A converse is a relation. Theorem 12 of [Suppes] p. 62.
|- Rel `'A
Theorem	cotr 3436	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 3437	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 3438	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 3439	Two ways of saying a relation is antisymmetric and reflexive. U.U.R is the field of a relation by relfld 3515.
|- ((R i^i `'R) = (I |` U.U.R) <-> A.x e. U.U.RA.y((xRy /\ yRx) <-> x = y))
Theorem	asymref2 3440	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 3441	Two ways of saying a relation is irreflexive. Definition of irreflexivity in [Schechter] p. 51.
|- ((R i^i I) = (/) <-> A.x -. xRx)
Theorem	soirri 3442	A strict order relation is irreflexive.
|- A e. V   &   |- R Or S   &   |- R (_ (S X. S)   =>   |- -. ARA
Theorem	sotri 3443	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 3444	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 3445	The converse of a class abstraction of ordered pairs.
|- `'{<.x, y>. | ph} = {<.y, x>. | ph}
Theorem	cnv0 3446	The converse of the empty set.
|- `'(/) = (/)
Theorem	cnvi 3447	The converse of the identity relation. Theorem 3.7(ii) of [Monk1] p. 36.
|- `'I = I
Theorem	op1sta 3448	Extract the first member of an ordered pair. (See op2nda 3452 to extract the second member, op1stb 2913 for an alternate version, and op1st 4085 for the preferred version..) (Contributed by Raph Levien, 4-Dec-2003.)
|- A e. V   =>   |- U.dom {<.A, B>.} = A
Theorem	cnvsn 3449	Converse of a singleton of an ordered pair.
|- A e. V   &   |- B e. V   =>   |- `'{<.A, B>.} = {<.B, A>.}
Theorem	rnsnop 3450	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 3451	Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 2913 to extract the first member, op2nda 3452 for an alternate version, and op2nd 4086 for the preferred version.)
|- A e. V   &   |- B e. V   =>   |- |^||^||^|`'{<.A, B>.} = B
Theorem	op2nda 3452	Extract the second member of an ordered pair. (See op1sta 3448 to extract the first member, op2ndb 3451 for an alternate version, and op2nd 4086 for the preferred version.)
|- A e. V   &   |- B e. V   =>   |- U.ran {<.A, B>.} = B
Theorem	elxp4 3453	Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 3454, elxp6 4102, and elxp7 4103.
|- (A e. (B X. C) <-> (A = <.U.dom { A}, U.ran { A}>. /\ (U.dom { A} e. B /\ U.ran { A} e. C)))
Theorem	elxp5 3454	Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 3453 when the double intersection does not create class existence problems (caused by int0 2547).
|- (A e. (B X. C) <-> (A = <.|^||^|A, U.ran { A}>. /\ (|^||^|A e. B /\ U.ran { A} e. C)))
Theorem	cnvun 3455	The converse of a union is the union of converses. Theorem 16 of [Suppes] p. 62.
|- `'(A u. B) = (`'A u. `'B)
Theorem	cnvin 3456	Distributive law for converse over intersection. Theorem 15 of [Suppes] p. 62.
|- `'(A i^i B) = (`'A i^i `'B)
Theorem	rnun 3457	Distributive law for range over union. Theorem 8 of [Suppes] p. 60.
|- ran ( A u. B) = (ran A u. ran B)
Theorem	rnin 3458	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 3459	The range of a union. Part of Exercise 8 of [Enderton] p. 41.
|- ran U. A = U_x e. A ran x
Theorem	imaun 3460	Distributive law for image over union. Theorem 35 of [Suppes] p. 65.
|- (A"(B u. C)) = ((A"B) u. (A"C))
Theorem	imaun2 3461	The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)
|- ((A u. B)"C) = ((A"C) u. (B"C))
Theorem	dminss 3462	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 3463	An upper bound for intersection with an image. Theorem 41 of [Suppes] p. 66.
|- ((R"A)