mmtheorems36 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 3501-3600 - Page 36 of 105

Type	Label	Description
Statement
Theorem	fncnv 3501	Single-rootedness (see funcnv 3497) 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 3502	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 3503	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 3504	The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 3497 for "single-rooted" definition.)
|- (A.f e. A (Fun `'f /\ A.g e. A (f (_ g \/ g (_ f)) -> Fun `'U.A)
Theorem	fun11uni 3505	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 3506	The intersection with a function is a function. Exercise 14(a) of [Enderton] p. 53.
|- (Fun F -> Fun (F i^i G))
Theorem	funres11 3507	The restriction of a one-to-one function is one-to-one.
|- (Fun `'F -> Fun `'(F |` A))
Theorem	funcnvres 3508	The converse of a restricted function.
|- (Fun `'F -> `'(F |` A) = (`'F |` (F"A)))
Theorem	cnvresid 3509	Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.)
|- `'(I |` A) = (I |` A)
Theorem	funcnvres2 3510	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 3511	The image of the pre-image of a function.
|- (Fun F -> (F"(`'F"A)) = (A i^i ran F))
Theorem	funimass1 3512	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 3513	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 3514	The image of a difference is the difference of images.
|- (Fun `'F -> (F"(A \ B)) = ((F"A) \ (F"B)))
Theorem	funimaexg 3515	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 3516	The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 2661. 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 3517	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 3518	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 3516.
|- 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 3519	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 3520	Existence of a composition when the first member is a function.
|- ((Fun A /\ B e. C) -> (A o. B) e. V)
Theorem	cofunex2g 3521	Existence of a composition when the second member is one-to-one.
|- ((A e. C /\ Fun `'B) -> (A o. B) e. V)
Theorem	fneq1 3522	Equality theorem for function predicate with domain.
|- (F = G -> (F Fn A <-> G Fn A))
Theorem	fneq2 3523	Equality theorem for function predicate with domain.
|- (A = B -> (F Fn A <-> F Fn B))
Theorem	hbfn 3524	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 3525	A function with domain is a function.
|- (F Fn A -> Fun F)
Theorem	fnrel 3526	A function with domain is a relation.
|- (F Fn A -> Rel F)
Theorem	fndm 3527	The domain of a function.
|- (F Fn A -> dom F = A)
Theorem	funfni 3528	Inference to convert a function and domain antecedent.
|- ((Fun F /\ B e. dom F) -> ph)   =>   |- ((F Fn A /\ B e. A) -> ph)
Theorem	fndmu 3529	A function has a unique domain.
|- ((F Fn A /\ F Fn B) -> A = B)
Theorem	fnbr 3530	The first argument of binary relation on a function belongs to the function's domain.
|- ((F Fn A /\ BFC) -> B e. A)
Theorem	fnop 3531	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 3532	There is exactly one value of a function.
|- ((F Fn A /\ B e. A) -> E!y BFy)
Theorem	fneu2 3533	There is exactly one value of a function.
|- ((F Fn A /\ B e. A) -> E!y<.B, y>. e. F)
Theorem	fnun 3534	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))
Theorem	fnco 3535	Composition of two functions.
|- ((F Fn A /\ G Fn B /\ ran G (_ A) -> (F o. G) Fn B)
Theorem	fnresdm 3536	A function does not change when restricted to its domain.
|- (F Fn A -> (F |` A) = F)
Theorem	fnresdisj 3537	A function restricted to a class disjoint with its domain is empty.
|- (F Fn A -> ((A i^i B) = (/) <-> (F |` B) = (/)))
Theorem	2elresin 3538	Membership in two functions restricted by each other's domain.
|- ((F Fn A /\ G Fn B) -> ((<.x, y>. e. F /\ <.x, z>. e. G) <-> (<.x, y>. e. (F |` (A i^i B)) /\ <.x, z>. e. (G |` (A i^i B)))))
Theorem	fnssresb 3539	Restriction of a function with a subclass of its domain.
|- (F Fn A -> ((F |` B) Fn B <-> B (_ A))
Theorem	fnssres 3540	Restriction of a function with a subclass of its domain.
|- ((F Fn A /\ B (_ A) -> (F |` B) Fn B)
Theorem	fnresin1 3541	Restriction of a function's domain with an intersection.
|- (F Fn A -> (F |` (A i^i B)) Fn (A i^i B))
Theorem	fnresin2 3542	Restriction of a function's domain with an intersection.
|- (F Fn A -> (F |` (B i^i A)) Fn (B i^i A))
Theorem	fnresi 3543	Functionality and domain of restricted identity.
|- (I |` A) Fn A
Theorem	fnima 3544	The image of a function's domain is its range.
|- (F Fn A -> (F"A) = ran F)
Theorem	fn0 3545	A function with empty domain is empty.
|- (F Fn (/) <-> F = (/))
Theorem	funimadisj 3546	A class that is disjoint with the domain of a function has an empty image under the function. (Contributed by FL, 24-Jan-2007.)
|- ((F Fn A /\ (A i^i C) = (/)) -> (F"C) = (/))
Theorem	fnex 3547	If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of funimaexg 3515.
|- ((F Fn A /\ A e. B) -> F e. V)
Theorem	funex 3548	If the domain of a function exists, so the function. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of fnex 3547. (Note: Any resemblance between F.U.N.E.X. and "Have You Any Eggs" is purely a coincidence originated by Swedish chefs.)
|- ((Fun F /\ dom F e. B) -> F e. V)
Theorem	opabex 3549	Existence of a function expressed as class of ordered pairs.
|- A e. V   &   |- (x e. A -> E*yph)   =>   |- {<.x, y>. | (x e. A /\ ph)} e. V
Theorem	opabex2 3550	Existence of a function expressed as class of ordered pairs.
|- A e. V   =>   |- {<.x, y>. | (x e. A /\ y = B)} e. V
Theorem	opabex2g 3551	Existence of a function expressed as class of ordered pairs.
|- (A e. C -> {<.x, y>. | (x e. A /\ y = B)} e. V)
Theorem	fopabex2 3552	Existence of a function expressed as class of ordered pairs.
|- A e. V   &   |- F = {<.x, y>. | (x e. A /\ y = B)}   =>   |- F e. V
Theorem	funrnex 3553	If the domain of a function exists, so does its range. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of funex 3548.
|- (dom F e. B -> (Fun F -> ran F e. V))
Theorem	zfrep6 3554	A version of the Axiom of Replacement. Normally ph would have free variables x and y. Axiom 6 of [Kunen] p. 12. The Separation Scheme ax-sep 2671 cannot be derived from this version and must be stated as a separate axiom in an axiom system (such as Kunen's) that uses this version in place of our ax-rep 2661.
|- (A.x e. z E!yph -> E.wA.x e. z E.y e. w ph)
Theorem	fnopabg 3555	Functionality and domain of an ordered-pair class abstraction.
|- F = {<.x, y>. | (x e. A /\ ph)}   =>   |- (A.x e. A E!yph <-> F Fn A)
Theorem	fnopab2g 3556	Functionality and domain of an ordered-pair class abstraction.
|- F = {<.x, y>. | (x e. A /\ y = B)}   =>   |- (A.x e. A B e. V <-> F Fn A)
Theorem	fnopab 3557	Functionality and domain of an ordered-pair class abstraction.
|- (x e. A -> E!yph)   &   |- F = {<.x, y>. | (x e. A /\ ph)}   =>   |- F Fn A
Theorem	fnopab2 3558	Functionality and domain of an ordered-pair class abstraction.
|- B e. V   &   |- F = {<.x, y>. | (x e. A /\ y = B)}   =>   |- F Fn A
Theorem	dmopab2 3559	Domain of an ordered-pair class abstraction that specifies a function.
|- B e. V   &   |- F = {<.x, y>. | (x e. A /\ y = B)}   =>   |- dom F = A
Theorem	feq1 3560	Equality theorem for functions.
|- (F = G -> (F:A-->B <-> G:A-->B))
Theorem	feq2 3561	Equality theorem for functions.
|- (A = B -> (F:A-->C <-> F:B-->C))
Theorem	feq3 3562	Equality theorem for functions.
|- (A = B -> (F:C-->A <-> F:C-->B))
Theorem	feq23 3563	Equality theorem for functions. (Contributed by FL, 14-Jul-2007.)
|- ((A = C