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	relssres 3401	Simplification law for restriction.
|- ((Rel A /\ dom A (_ B) -> (A |` B) = A)
Theorem	resdm 3402	A relation restricted to its domain equals itself.
|- (Rel A -> (A |` dom A) = A)
Theorem	resexg 3403	The restriction of a set is a set.
|- (A e. C -> (A |` B) e. V)
Theorem	resopab 3404	Restriction of a class abstraction of ordered pairs.
|- ({<.x, y>. | ph} |` A) = {<.x, y>. | (x e. A /\ ph)}
Theorem	resiexg 3405	The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 3588).
|- (A e. B -> (I |` A) e. V)
Theorem	iss 3406	A subclass of the identity function is the identity function restricted to its domain.
|- (A (_ I <-> A = (I |` dom A))
Theorem	resopab2 3407	Restriction of a class abstraction of ordered pairs.
|- (A (_ B -> ({<.x, y>. | (x e. B /\ ph)} |` A) = {<.x, y>. | (x e. A /\ ph)})
Theorem	dmresi 3408	The domain of a restricted identity function.
|- dom ( I |` A) = A
Theorem	resid 3409	Any relation restricted to the universe is itself.
|- (Rel A -> (A |` V) = A)
Theorem	imaeq1 3410	Equality theorem for image.
|- (A = B -> (A"C) = (B"C))
Theorem	imaeq2 3411	Equality theorem for image.
|- (A = B -> (C"A) = (C"B))
Theorem	imaeq1d 3412	Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
|- (ph -> A = B)   =>   |- (ph -> (A"C) = (B"C))
Theorem	imaeq2d 3413	Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
|- (ph -> A = B)   =>   |- (ph -> (C"A) = (C"B))
Theorem	dfima2 3414	Alternate definition of image. Compare definition (d) of [Enderton] p. 44.
|- (A"B) = {y | E.x e. B xAy}
Theorem	dfima3 3415	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 3416	Membership in an image. Theorem 34 of [Suppes] p. 65.
|- (A e. D -> (A e. (B"C) <-> E.x e. C xBA))
Theorem	elima 3417	Membership in an image. Theorem 34 of [Suppes] p. 65.
|- A e. V   =>   |- (A e. (B"C) <-> E.x e. C xBA)
Theorem	elima2 3418	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 3419	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 3420	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 3421	Deduction version of bound-variable hypothesis builder hbima 3420. (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 3422	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 3423	The image of the domain of a class is the range of the class.
|- (A"dom A) = ran A
Theorem	imassrn 3424	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 3425	The image of a set is a set. Theorem 3.17 of [Monk1] p. 39.
|- (A e. C -> (A"B) e. V)
Theorem	imai 3426	Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38.
|- (I"A) = A
Theorem	rnresi 3427	The range of the restricted identity function.
|- ran ( I |` A) = A
Theorem	resiima 3428	The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)
|- (B (_ A -> ((I |` A)"B) = B)
Theorem	ima0 3429	Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38.
|- (A"(/)) = (/)
Theorem	0ima 3430	Image under the empty relation. (Contributed by FL, 11-Jan-2007.)
|- ((/)"A) = (/)
Theorem	imadisj 3431	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 3432	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 3433	The image of a singleton.
|- (A e. B -> (R"{A}) = {y | ARy})
Theorem	relimasn 3434	The image of a singleton.
|- (Rel R -> (R"{A}) = {y | ARy})
Theorem	elimasn 3435	Membership in an image of a singleton.
|- B e. V   &   |- C e. V   =>   |- (C e. (A"{B}) <-> <.B, C>. e. A)
Theorem	elimasng 3436	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 3437	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 3207, which was based on the idea in Quine's definition.
|- {x | E.y(F"{x}) = {y}} = {x | E!y xFy}
Theorem	eliniseg 3438	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 3439	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 3440	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 3441	Subset theorem for image.
|- (A (_ B -> (A"C) (_ (B"C))
Theorem	imass2 3442	Subset theorem for image. Exercise 22(a) of [Enderton] p. 53.
|- (A (_ B -> (C"A) (_ (C"B))
Theorem	ndmima 3443	The image of a singleton outside the domain is empty.
|- (-. A e. dom B -> (B"{A}) = (/))
Theorem	relcnv 3444	A converse is a relation. Theorem 12 of [Suppes] p. 62.
|- Rel `'A
Theorem	cotr 3445	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 3446	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 3447	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 3448	Two ways of saying a relation is antisymmetric and reflexive. U.U.R is the field of a relation by relfld 3524.
|- ((R i^i `'R) = (I |` U.U.R) <-> A.x e. U.U.RA.y((xRy /\ yRx) <-> x = y))
Theorem	asymref2 3449	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 3450	Two ways of saying a relation is irreflexive. Definition of irreflexivity in [Schechter] p. 51.
|- ((R i^i I) = (/) <-> A.x -. xRx)
Theorem	soirri 3451	A strict order relation is irreflexive.
|- A e. V   &   |- R Or S   &   |- R (_ (S X. S)   =>   |- -. ARA
Theorem	sotri 3452	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 3453	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 3454	The converse of a class abstraction of ordered pairs.
|- `'{<.x, y>. | ph} = {<.y, x>. | ph}
Theorem	cnv0 3455	The converse of the empty set.
|- `'(/) = (/)
Theorem	cnvi 3456	The converse of the identity relation. Theorem 3.7(ii) of [Monk1] p. 36.
|- `'I = I
Theorem	op1sta 3457	Extract the first member of an ordered pair. (See op2nda 3461 to extract the second member, op1stb 2922 for an alternate version, and op1st 4094 for the preferred version..) (Contributed by Raph Levien, 4-Dec-2003.)
|- A e. V   =>   |- U.dom {<.A, B>.} = A
Theorem	cnvsn 3458	Converse of a singleton of an ordered pair.
|- A e. V   &   |- B e. V   =>   |- `'{<.A, B>.} = {<.B, A>.}
Theorem	rnsnop 3459	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 3460	Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 2922 to extract the first member, op2nda 3461 for an alternate version, and op2nd 4095 for the preferred version.)
|- A e. V   &   |- B e. V   =>   |- |^||^||^|`'{<.A, B>.} = B
Theorem	op2nda 3461	Extract the second member of an ordered pair. (See op1sta 3457 to extract the first member, op2ndb 3460 for an alternate version, and op2nd 4095 for the preferred version.)
|- A e. V   &   |- B e. V   =>   |- U.ran {<.A, B>.} = B
Theorem	elxp4 3462	Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 3463, elxp6 4111, and elxp7 4112.
|- (A e. (B X. C) <-> (A = <.U.dom { A}, U.ran { A}>. /\ (U.dom { A} e. B /\ U.ran { A} e. C)))
Theorem	elxp5 3463	Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 3462 when the double intersection does not create class existence problems (caused by int0 2554).
|- (A e. (B X. C) <-> (A = <.|^||^|A, U.ran {