mmtheorems37 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 3601-3700 - Page 37 of 123

Type	Label	Description
Statement
Theorem	coundir 3601	Class composition distributes over union.
|- ((A u. B) o. C) = ((A o. C) u. (B o. C))
Theorem	cores 3602	Restricted first member of a class composition.
|- (ran B (_ C -> ((A |` C) o. B) = (A o. B))
Theorem	resco 3603	Associative law for the restriction of a composition.
|- ((A o. B) |` C) = (A o. (B |` C))
Theorem	imaco 3604	Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.)
|- ((A o. B)"C) = (A"(B"C))
Theorem	rnco 3605	The range of the composition of two classes.
|- ran ( A o. B) = ran ( A |` ran B)
Theorem	rnco2 3606	The range of the composition of two classes.
|- ran ( A o. B) = (A"ran B)
Theorem	dmco 3607	The domain of a composition. Exercise 27 of [Enderton] p. 53.
|- dom ( A o. B) = (`'B"dom A)
Theorem	coiun 3608	Composition with an indexed union.
|- (A o. U_x e. C B) = U_x e. C (A o. B)
Theorem	cocnvcnv1 3609	A composition is not affected by a double converse of its first argument.
|- (`'`'A o. B) = (A o. B)
Theorem	cocnvcnv2 3610	A composition is not affected by a double converse of its second argument.
|- (A o. `'`'B) = (A o. B)
Theorem	cores2 3611	Absorption of a reverse (preimage) restriction of the second member of a class composition.
|- (dom A (_ C -> (A o. `'(`'B |` C)) = (A o. B))
Theorem	co02 3612	Composition with the empty set. Theorem 20 of [Suppes] p. 63.
|- (A o. (/)) = (/)
Theorem	co01 3613	Composition with the empty set.
|- ((/) o. A) = (/)
Theorem	coi1 3614	Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36.
|- (Rel A -> (A o. I) = A)
Theorem	coi2 3615	Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36.
|- (Rel A -> (I o. A) = A)
Theorem	coass 3616	Associative law for class composition. Theorem 27 of [Suppes] p. 64. Also Exercise 21 of [Enderton] p. 53. Interestingly, this law holds for any classes whatsoever, not just functions or even relations.
|- ((A o. B) o. C) = (A o. (B o. C))
Theorem	relssdmrn 3617	A relation is included in the cross product of its domain and range. Exercise 4.12(t) of [Mendelson] p. 235.
|- (Rel A -> A (_ (dom A X. ran A))
Theorem	relrelss 3618	Two ways to describe the structure of a two-place operation.
|- ((Rel A /\ Rel dom A) <-> A (_ ((V X. V) X. V))
Theorem	unielrel 3619	The membership relation for a relation is inherited by class union.
|- ((Rel R /\ A e. R) -> U.A e. U.R)
Theorem	relfld 3620	The double union of a relation is its field.
|- (Rel R -> U.U.R = (dom R u. ran R))
Theorem	unidmrn 3621	The double union of the converse of a class is its field.
|- U.U.`'A = (dom A u. ran A)
Theorem	unixp 3622	The double class union of a non-empty cross product is the union of it members.
|- ((A X. B) =/= (/) -> U.U.(A X. B) = (A u. B))
Theorem	unixp0 3623	A cross product is empty iff its union is empty.
|- ((A X. B) = (/) <-> U.(A X. B) = (/))
Theorem	cnvexg 3624	The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26.
|- (A e. B -> `'A e. V)
Theorem	cnvex 3625	The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26.
|- A e. V   =>   |- `'A e. V
Theorem	relcnvexb 3626	A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.)
|- (Rel R -> (R e. V <-> `'R e. V))
Theorem	cnvpo 3627	The converse of a partial order relation is a partial order relation.
|- (R Po A <-> `'R Po A)
Theorem	cnvso 3628	The converse of a strict order relation is a strict order relation.
|- (R Or A <-> `'R Or A)
Theorem	coexg 3629	The composition of two sets is a set.
|- ((A e. C /\ B e. D) -> (A o. B) e. V)
Theorem	coex 3630	The composition of two sets is a set.
|- A e. V   &   |- B e. V   =>   |- (A o. B) e. V
Theorem	dffun2 3631	Alternate definition of a function.
|- (Fun A <-> (Rel A /\ A.xA.yA.z((xAy /\ xAz) -> y = z)))
Theorem	dffun3 3632	Alternate definition of function.
|- (Fun A <-> (Rel A /\ A.xE.zA.y(xAy -> y = z)))
Theorem	dffun4 3633	Alternate definition of a function. Definition 6.4(4) of [TakeutiZaring] p. 24.
|- (Fun A <-> (Rel A /\ A.xA.yA.z((<.x, y>. e. A /\ <.x, z>. e. A) -> y = z)))
Theorem	dffun5 3634	Alternate definition of function.
|- (Fun A <-> (Rel A /\ A.xE.zA.y(<.x, y>. e. A -> y = z)))
Theorem	dffun6f 3635	Definition of function, 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)   =>   |- (Fun A <-> (Rel A /\ A.xE*y xAy))
Theorem	dffun6 3636	Alternate definition of a function using "at most one" notation.
|- (Fun A <-> (Rel A /\ A.xE*y xAy))
Theorem	funmo 3637	A function has at most one value for each argument.
|- (Fun A -> E*y xAy)
Theorem	funrel 3638	A function is a relation.
|- (Fun A -> Rel A)
Theorem	funss 3639	Subclass theorem for function predicate.
|- (A (_ B -> (Fun B -> Fun A))
Theorem	funeq 3640	Equality theorem for function predicate.
|- (A = B -> (Fun A <-> Fun B))
Theorem	hbfun 3641	Bound-variable hypothesis builder for a function.
|- (y e. F -> A.x y e. F)   =>   |- (Fun F -> A.xFun F)
Theorem	funeu 3642	There is exactly one value of a function.
|- ((Fun F /\ xFy) -> E!y xFy)
Theorem	funeu2 3643	There is exactly one value of a function.
|- ((Fun F /\ <.x, y>. e. F) -> E!y<.x, y>. e. F)
Theorem	dffun7 3644	Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. (Enderton's definition is ambiguous because "there is only one" could mean either "there is at most one" or "there is exactly one." However, dffun8 3645 shows that it doesn't matter which meaning we pick.)
|- (Fun A <-> (Rel A /\ A.x e. dom AE*y xAy))
Theorem	dffun8 3645	Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 3644.
|- (Fun A <-> (Rel A /\ A.x e. dom AE!y xAy))
Theorem	dffun9 3646	Alternate definition of a function.
|- (Fun A <-> (Rel A /\ A.x e. dom AE*y(y e. ran A /\ xAy)))
Theorem	funfn 3647	An equivalence for the function predicate.
|- (Fun A <-> A Fn dom A)
Theorem	funsn 3648	A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65.
|- A e. V   &   |- B e. V   =>   |- Fun {<.A, B>.}
Theorem	fun0 3649	The empty set is a function. Theorem 10.3 of [Quine] p. 65.
|- Fun (/)
Theorem	funi 3650	The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65.
|- Fun I
Theorem	nfunv 3651	The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.)
|- -. Fun V
Theorem	funopg 3652	A Kuratowski ordered pair is a function only if its components are equal.
|- ((B e. C /\ Fun <.A, B>.) -> A = B)
Theorem	funopab 3653	A class of ordered pairs is a function when there is at most one second member for each pair.
|- (Fun {<.x, y>. | ph} <-> A.xE*yph)
Theorem	funopabeq 3654	A class of ordered pairs of values is a function.
|- Fun {<.x, y>. | y = A}
Theorem	funco 3655	The composition of two functions is a function. Exercise 29 of [TakeutiZaring] p. 25.
|- ((Fun F /\ Fun G) -> Fun (F o. G))
Theorem	funres 3656	A restriction of a function is a function. Compare Exercise 18 of [TakeutiZaring] p. 25.
|- (Fun F -> Fun (F |` A))
Theorem	funssres 3657	The restriction of a function to the domain of a subclass equals the subclass.
|- ((Fun F /\ G (_ F) -> (F |` dom G) = G)
Theorem	fun2ssres 3658	Equality of restrictions of a function and a subclass.
|- ((Fun F /\ G (_ F /\ A (_ dom G) -> (F |` A) = (G |` A))
Theorem	funun 3659	The union of functions with disjoint domains is a function. Theorem 4.6 of [Monk1] p. 43.
|- (((Fun F /\ Fun G) /\ (dom F i^i dom G) = (/)) -> Fun (F u. G))
Theorem	funcnvcnv 3660	The double converse of a function is a function.
|- (Fun A -> Fun `'`'A)
Theorem	funcnv2 3661	A simpler equivalence for single-rooted (see funcnv 3662).
|- (Fun `'A <-> A.yE*x xAy)
Theorem	funcnv 3662	The converse of a class is a function iff the class is single-rooted, which means that for any y in the range of A there is at most one x such that xAy. Definition of single-rooted in [Enderton] p. 43. See funcnv2 3661 for a simpler version.
|- (Fun `'A <-> A.y e. ran AE*x xAy)
Theorem	funcnv3 3663	A condition showing a class is single-rooted. (See funcnv 3662).
|- (Fun `'A <-> A.y e. ran AE!x e. dom A xAy)
Theorem	fun2cnv 3664	The double converse of a class is a function iff the class is single-valued. Each side is equivalent to Definition 6.4(2) of [TakeutiZaring] p. 23, who use the notation "Un(A)" for single-valued. Note that A is not necessarily a function.
|- (Fun `'`'A <-> A.xE*y xAy)
Theorem	svrelfun 3665	A single-valued relation is a function. (See fun2cnv 3664 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24.
|- (Fun A <-> (Rel A /\ Fun `'`'A))
Theorem	fncnv 3666	Single-rootedness (see funcnv 3662) of a class cut down by a cross product.
|- (`'(R i^i (A X. B)) Fn B <-> A.y e. B E!x e. A xRy)
Theorem	fun11 3667	Two ways of stating that A is one-to-one (but not necessarily a function). Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un2 (A)" for one-to-one (but not necessarily a function).
|- ((Fun `'`'A /\ Fun `'A) <-> A.xA.yA.zA.w((xAy /\ zAw) -> (x = z <-> y = w)))
Theorem	fununi 3668	The union of a chain (with respect to inclusion) of functions is a function.
|- (A.f e. A (Fun f /\ A.g e. A (f (_ g \/ g (_ f)) -> Fun U.A)
Theorem	funcnvuni 3669	The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 3662 for "single-rooted" definition.)
|- (A.f e. A (Fun `'f /\ A.g e. A (f (_ g \/ g (_ f)) -> Fun `'U.A)
Theorem	fun11uni 3670	The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function.
|- (A.f e. A ((Fun f /\ Fun `'f) /\ A.g e. A (f (_ g \/ g (_ f)) -> (Fun U.A /\ Fun `'U.A))
Theorem	funin 3671	The intersection with a function is a function. Exercise 14(a) of [Enderton] p. 53.
|- (Fun F -> Fun (F i^i G))
Theorem	funres11 3672	The restriction of a one-to-one function is one-to-one.
|- (Fun `'F -> Fun `'(F |` A))
Theorem	funcnvres 3673	The converse of a restricted function.
|- (Fun `'F -> `'(F |` A) = (`'F |` (F"A)))
Theorem	cnvresid 3674	Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.)
|- `'(I |` A) = (I |` A)
Theorem	funcnvres2 3675	The converse of a restriction of the converse of a function equals the function restricted to the image of its converse.
|- (Fun F -> `'(`'F |` A) = (F |` (`'F"A)))
Theorem	funimacnv 3676	The image of the pre-image of a function.
|- (Fun F -> (F"(`'F"A)) = (A i^i ran F))
Theorem	funimass1 3677	A kind of contraposition law that infers a subclass of an image from a pre-image subclass.
|- ((Fun F /\ A (_ ran F) -> ((`'F"A) (_ B -> A (_ (F"B)))
Theorem	funimass2 3678	A kind of contraposition law that infers an image subclass from a subclass of a pre-image.
|- ((Fun F /\ A (_ (`'F"B)) -> (F"A) (_ B)
Theorem	imadif 3679	The image of a difference is the difference of images.
|- (Fun `'F -> (F"(A \ B)) = ((F"A) \ (F"B)))
Theorem	imain 3680	The image of an intersection is the intersection of images. (Contributed by Paul Chapman, 11-Apr-2009.)
|- (Fun `'F -> (F"(A i^i B)) = ((F"A) i^i (F"B)))
Theorem	funimaexg 3681	Axiom of Replacement using abbreviations. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29.
|- ((Fun A /\ B e. C) -> (A"B) e. V)
Theorem	funimaex 3682	The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 2767. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29.
|- B e. V   =>   |- (Fun A -> (A"B) e. V)
Theorem	isarep1 3683	Part of a study of the Axiom of Replacement used by the Isabelle prover. The object PrimReplace is apparently the image of the function encoded by ph(x, y) i.e. the class ({<.x, y>. | ph}"A). If so, we can prove Isabelle's "Axiom of Replacement" conclusion without using the Axiom of Replacement, for which I (N. Megill) currently have no explanation.
|- (b e. ({<.x, y>. | ph}"A) <-> E.x e. A [b / y]ph)
Theorem	isarep2 3684	Part of a study of the Axiom of Replacement used by the Isabelle prover. In Isabelle, the sethood of PrimReplace is apparently postulated implicitly by its type signature "[ i, [ i, i ] => o ] => i", which automatically asserts that it is a set without using any axioms. To prove that it is a set in Metamath, we need the hypotheses of Isabelle's "Axiom of Replacement" as well as the Axiom of Replacement in the form funimaex 3682.
|- A e. V   &   |- A.x e. A A.yA.z((ph /\ [z / y]ph) -> y = z)   =>   |- E.w w = ({<.x, y>. | ph}"A)
Theorem	resfunexg 3685	The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28.
|- ((Fun A /\ B e. C) -> (A |` B) e. V)
Theorem	cofunexg 3686	Existence of a composition when the first member is a function.
|- ((Fun A /\ B e. C) -> (A o. B) e. V)
Theorem	cofunex2g 3687	Existence of a composition when the second member is one-to-one.
|- ((A e. C /\ Fun `'B) -> (A o. B) e. V)
Theorem	fneq1 3688	Equality theorem for function predicate with domain.
|- (F = G -> (F Fn A <-> G Fn A))
Theorem	fneq2 3689	Equality theorem for function predicate with domain.
|- (A = B -> (F Fn A <-> F Fn B))
Theorem	hbfn 3690	Bound-variable hypothesis builder for a function with domain.
|- (y e. F -> A.x y e. F)   &   |- (y e. A -> A.x y e. A)   =>   |- (F Fn A -> A.x F Fn A)
Theorem	fnfun 3691	A function with domain is a function.
|- (F Fn A -> Fun F)
Theorem	fnrel 3692	A function with domain is a relation.
|- (F Fn A -> Rel F)
Theorem	fndm 3693	The domain of a function.
|- (F Fn A -> dom F = A)
Theorem	funfni 3694	Inference to convert a function and domain antecedent.
|- ((Fun F /\ B e. dom F) -> ph)   =>   |- ((F Fn A /\ B e. A) -> ph)
Theorem	fndmu 3695	A function has a unique domain.
|- ((F Fn A /\ F Fn B) -> A = B)
Theorem	fnbr 3696	The first argument of binary relation on a function belongs to the function's domain.
|- ((F Fn A /\ BFC) -> B e. A)
Theorem	fnop 3697	The first argument of an ordered pair in a function belongs to the function's domain.
|- ((F Fn A /\ <.B, C>. e. F) -> B e. A)
Theorem	fneu 3698	There is exactly one value of a function.
|- ((F Fn A /\ B e. A) -> E!y BFy)
Theorem	fneu2 3699	There is exactly one value of a function.
|- ((F Fn A /\ B e. A) -> E!y<.B, y>. e. F)
Theorem	fnun 3700	The union of two functions with disjoint domains.
|- (((F Fn A /\ G Fn B) /\ (A i^i B) = (/)) -> (F u. G) Fn (A u. B))