mmtheorems36 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 3501-3600 - Page 36 of 107

Type	Label	Description
Statement
Theorem	cocnvcnv2 3501	A composition is not affected by a double converse of its second argument.
⊢ (A ∘ ◡◡B) = (A ∘ B)
Theorem	cores2 3502	Absorption of a reverse (preimage) restriction of the second member of a class composition.
⊢ (dom A ⊆ C → (A ∘ ◡(◡B ↾ C)) = (A ∘ B))
Theorem	co02 3503	Composition with the empty set. Theorem 20 of [Suppes] p. 63.
⊢ (A ∘ ∅) = ∅
Theorem	co01 3504	Composition with the empty set.
⊢ (∅ ∘ A) = ∅
Theorem	coi1 3505	Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36.
⊢ (Rel A → (A ∘ I) = A)
Theorem	coi2 3506	Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36.
⊢ (Rel A → (I ∘ A) = A)
Theorem	coass 3507	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 ∘ B) ∘ C) = (A ∘ (B ∘ C))
Theorem	relssdr 3508	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 × ran A))
Theorem	unielrel 3509	The membership relation for a relation is inherited by class union.
⊢ ((Rel R ⋀ A ∈ R) → ∪A ∈ ∪R)
Theorem	relfld 3510	The double union of a relation is its field.
⊢ (Rel R → ∪∪R = (dom R ∪ ran R))
Theorem	unidmrn 3511	The double union of the converse of a class is its field.
⊢ ∪∪◡A = (dom A ∪ ran A)
Theorem	unixp 3512	The double class union of a non-empty cross product is the union of it members.
⊢ ((A × B) ≠ ∅ → ∪∪(A × B) = (A ∪ B))
Theorem	unixp0 3513	A cross product is empty iff its union is empty.
⊢ ((A × B) = ∅ ↔ ∪(A × B) = ∅)
Theorem	cnvexg 3514	The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26.
⊢ (A ∈ B → ◡A ∈ V)
Theorem	cnvex 3515	The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26.
⊢ A ∈ V    ⇒   ⊢ ◡A ∈ V
Theorem	relcnvexb 3516	A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.)
⊢ (Rel R → (R ∈ V ↔ ◡R ∈ V))
Theorem	cnvpo 3517	The converse of a partial order relation is a partial order relation.
⊢ (R Po A ↔ ◡R Po A)
Theorem	cnvso 3518	The converse of a strict order relation is a strict order relation.
⊢ (R Or A ↔ ◡R Or A)
Theorem	coexg 3519	The composition of two sets is a set.
⊢ ((A ∈ C ⋀ B ∈ D) → (A ∘ B) ∈ V)
Theorem	coex 3520	The composition of two sets is a set.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ (A ∘ B) ∈ V
Theorem	dffun2 3521	Alternate definition of a function.
⊢ (Fun A ↔ (Rel A ⋀ ∀x∀y∀z((xAy ⋀ xAz) → y = z)))
Theorem	dffun3 3522	Alternate definition of function.
⊢ (Fun A ↔ (Rel A ⋀ ∀x∃z∀y(xAy → y = z)))
Theorem	dffun4 3523	Alternate definition of a function. Definition 6.4(4) of [TakeutiZaring] p. 24.
⊢ (Fun A ↔ (Rel A ⋀ ∀x∀y∀z((⟨x, y⟩ ∈ A ⋀ ⟨x, z⟩ ∈ A) → y = z)))
Theorem	dffun5 3524	Alternate definition of function.
⊢ (Fun A ↔ (Rel A ⋀ ∀x∃z∀y(⟨x, y⟩ ∈ A → y = z)))
Theorem	dffunmof 3525	Definition of function, using bound-variable hypotheses instead of distinct variable conditions.
⊢ (z ∈ A → ∀x z ∈ A)    &   ⊢ (z ∈ A → ∀y z ∈ A)    ⇒   ⊢ (Fun A ↔ (Rel A ⋀ ∀x∃*y xAy))
Theorem	dffunmo 3526	Alternate definition of a function using "at most one" notation.
⊢ (Fun A ↔ (Rel A ⋀ ∀x∃*y xAy))
Theorem	funmo 3527	A function has at most one value for each argument.
⊢ (Fun A → ∃*y xAy)
Theorem	funrel 3528	A function is a relation.
⊢ (Fun A → Rel A)
Theorem	funss 3529	Subclass theorem for function predicate.
⊢ (A ⊆ B → (Fun B → Fun A))
Theorem	funeq 3530	Equality theorem for function predicate.
⊢ (A = B → (Fun A ↔ Fun B))
Theorem	hbfun 3531	Bound-variable hypothesis builder for a function.
⊢ (y ∈ F → ∀x y ∈ F)    ⇒   ⊢ (Fun F → ∀xFun F)
Theorem	funeu 3532	There is exactly one value of a function.
⊢ ((Fun F ⋀ xFy) → ∃!y xFy)
Theorem	funeu2 3533	There is exactly one value of a function.
⊢ ((Fun F ⋀ ⟨x, y⟩ ∈ F) → ∃!y⟨x, y⟩ ∈ F)
Theorem	dffun6 3534	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, dffun7 3535 shows that it doesn't matter which meaning we pick.)
⊢ (Fun A ↔ (Rel A ⋀ ∀x ∈ dom A∃*y xAy))
Theorem	dffun7 3535	Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun6 3534.
⊢ (Fun A ↔ (Rel A ⋀ ∀x ∈ dom A∃!y xAy))
Theorem	dffun8 3536	Alternate definition of a function.
⊢ (Fun A ↔ (Rel A ⋀ ∀x ∈ dom A∃*y(y ∈ ran A ⋀ xAy)))
Theorem	funfn 3537	An equivalence for the function predicate.
⊢ (Fun A ↔ A Fn dom A)
Theorem	funsn 3538	A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ Fun {⟨A, B⟩}
Theorem	fun0 3539	The empty set is a function. Theorem 10.3 of [Quine] p. 65.
⊢ Fun ∅
Theorem	funi 3540	The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65.
⊢ Fun I
Theorem	nfunv 3541	The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.)
⊢ ¬ Fun V
Theorem	funopg 3542	A Kuratowski ordered pair is a function only if its components are equal.
⊢ ((B ∈ C ⋀ Fun ⟨A, B⟩) → A = B)
Theorem	funopab 3543	A class of ordered pairs is a function when there is at most one second member for each pair.
⊢ (Fun {⟨x, y⟩∣φ} ↔ ∀x∃*yφ)
Theorem	funopabeq 3544	A class of ordered pairs of values is a function.
⊢ Fun {⟨x, y⟩∣y = A}
Theorem	funco 3545	The composition of two functions is a function. Exercise 29 of [TakeutiZaring] p. 25.
⊢ ((Fun F ⋀ Fun G) → Fun (F ∘ G))
Theorem	funres 3546	A restriction of a function is a function. Compare Exercise 18 of [TakeutiZaring] p. 25.
⊢ (Fun F → Fun (F ↾ A))
Theorem	funssres 3547	The restriction of a function to the domain of a subclass equals the subclass.
⊢ ((Fun F ⋀ G ⊆ F) → (F ↾ dom G) = G)
Theorem	fun2ssres 3548	Equality of restrictions of a function and a subclass.
⊢ ((Fun F ⋀ G ⊆ F ⋀ A ⊆ dom G) → (F ↾ A) = (G ↾ A))
Theorem	funun 3549	The union of functions with disjoint domains is a function. Theorem 4.6 of [Monk1] p. 43.
⊢ (((Fun F ⋀ Fun G) ⋀ (dom F ∩ dom G) = ∅) → Fun (F ∪ G))
Theorem	funcnvcnv 3550	The double converse of a function is a function.
⊢ (Fun A → Fun ◡◡A)
Theorem	funcnv2 3551	A simpler equivalence for single-rooted (see funcnv 3552).
⊢ (Fun ◡A ↔ ∀y∃*x xAy)
Theorem	funcnv 3552	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 3551 for a simpler version.
⊢ (Fun ◡A ↔ ∀y ∈ ran A∃*x xAy)
Theorem	funcnv3 3553	A condition showing a class is single-rooted. (See funcnv 3552).
⊢ (Fun ◡A ↔ ∀y ∈ ran A∃!x ∈ dom A xAy)
Theorem	fun2cnv 3554	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 ↔ ∀x∃*y xAy)
Theorem	svrelfun 3555	A single-valued relation is a function. (See fun2cnv 3554 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24.
⊢ (Fun A ↔ (Rel A ⋀ Fun ◡◡A))
Theorem	fncnv 3556	Single-rootedness (see funcnv 3552) of a class cut down by a cross product.
⊢ (◡(R ∩ (A × B)) Fn B ↔ ∀y ∈ B ∃!x ∈ A xRy)
Theorem	fun11 3557	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) ↔ ∀x∀y∀z∀w((xAy ⋀ zAw) → (x = z ↔ y = w)))
Theorem	fununi 3558	The union of a chain (with respect to inclusion) of functions is a function.
⊢ (∀f ∈ A (Fun f ⋀ ∀g ∈ A (f ⊆ g ⋁ g ⊆ f)) → Fun ∪A)
Theorem	funcnvuni 3559	The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 3552 for "single-rooted" definition.)
⊢ (∀f ∈ A (Fun ◡f ⋀ ∀g ∈ A (f ⊆ g ⋁ g ⊆ f)) → Fun ◡∪A)
Theorem	fun11uni 3560	The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function.
⊢ (∀f ∈ A ((Fun f ⋀ Fun ◡f) ⋀ ∀g ∈ A (f ⊆ g ⋁ g ⊆ f)) → (Fun ∪A ⋀ Fun ◡∪A))
Theorem	funin 3561	The intersection with a function is a function. Exercise 14(a) of [Enderton] p. 53.
⊢ (Fun F → Fun (F ∩ G))
Theorem	funres11 3562	The restriction of a one-to-one function is one-to-one.
⊢ (Fun ◡F → Fun ◡(F ↾ A))
Theorem	funcnvres 3563	The converse of a restricted function.
⊢ (Fun ◡F → ◡(F ↾ A) = (◡F ↾ (F “ A)))
Theorem	cnvresid 3564	Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.)
⊢ ◡(I ↾ A) = (I ↾ A)
Theorem	funcnvres2 3565	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 3566	The image of the pre-image of a function.
⊢ (Fun F → (F “ (◡F “ A)) = (A ∩ ran F))
Theorem	funimass1 3567	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 3568	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 3569	The image of a difference is the difference of images.
⊢ (Fun ◡F → (F “ (A ∖ B)) = ((F “ A) ∖ (F “ B)))
Theorem	funimaexg 3570	Axiom of Replacement using abbreviations. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29.
⊢ ((Fun A ⋀ B ∈ C) → (A “ B) ∈ V)
Theorem	funimaex 3571	The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 2689. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29.
⊢ B ∈ V    ⇒   ⊢ (Fun A → (A “ B) ∈ V)
Theorem	isarep1 3572	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 φ(x, y) i.e. the class ({⟨x, y⟩∣φ} “ 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 ∈ ({⟨x, y⟩∣φ} “ A) ↔ ∃x ∈ A [b / y]φ)
Theorem	isarep2 3573	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 3571.
⊢ A ∈ V    &   ⊢ ∀x ∈ A ∀y∀z((φ ⋀ [z / y]φ) → y = z)    ⇒   ⊢ ∃w w = ({⟨x, y⟩∣φ} “ A)
Theorem	resfunexg 3574	The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28.
⊢ ((Fun A ⋀ B ∈ C) → (A ↾ B) ∈ V)
Theorem	cofunexg 3575	Existence of a composition when the first member is a function.
⊢ ((Fun A ⋀ B ∈ C) → (A ∘ B) ∈ V)
Theorem	cofunex2g 3576	Existence of a composition when the second member is one-to-one.
⊢ ((A ∈ C ⋀ Fun ◡B) → (A ∘ B) ∈ V)
Theorem	fneq1 3577	Equality theorem for function predicate with domain.
⊢ (F = G → (F Fn A ↔ G Fn A))
Theorem	fneq2 3578	Equality theorem for function predicate with domain.
⊢ (A = B → (F Fn A ↔ F Fn B))
Theorem	hbfn 3579	Bound-variable hypothesis builder for a function with domain.
⊢ (y ∈ F → ∀x y ∈ F)    &   ⊢ (y ∈ A → ∀x y ∈ A)    ⇒   ⊢ (F Fn A → ∀x F Fn A)
Theorem	fnfun 3580	A function with domain is a function.
⊢ (F Fn A → Fun F)
Theorem	fnrel 3581	A function with domain is a relation.
⊢ (F Fn A → Rel F)
Theorem	fndm 3582	The domain of a function.
⊢ (F Fn A → dom F = A)
Theorem	funfni 3583	Inference to convert a function and domain antecedent.
⊢ ((Fun F ⋀ B ∈ dom F) → φ)    ⇒   ⊢ ((F Fn A ⋀ B ∈ A) → φ)
Theorem	fndmu 3584	A function has a unique domain.
⊢ ((F Fn A ⋀ F Fn B) → A = B)
Theorem	fnbr 3585	The first argument of binary relation on a function belongs to the function's domain.
⊢ ((F Fn A ⋀ BFC) → B ∈ A)
Theorem	fnop 3586	The first argument of an ordered pair in a function belongs to the function's domain.
⊢ ((F Fn A ⋀ ⟨B, C⟩ ∈ F) → B ∈ A)
Theorem	fneu 3587	There is exactly one value of a function.
⊢ ((F Fn A ⋀ B ∈ A) → ∃!y BFy)
Theorem	fneu2 3588	There is exactly one value of a function.
⊢ ((F Fn A ⋀ B ∈ A) → ∃!y⟨B, y⟩ ∈ F)
Theorem	fnun 3589	The union of two functions with disjoint domains.
⊢ (((F Fn A ⋀ G Fn B) ⋀ (A ∩ B) = ∅) → (F ∪ G) Fn (A ∪ B))
Theorem	fnco 3590	Composition of two functions.
⊢ ((F Fn A ⋀ G Fn B ⋀ ran G ⊆ A) → (F ∘ G) Fn B)
Theorem	fnresdm 3591	A function does not change when restricted to its domain.
⊢ (F Fn A → (F ↾ A) = F)
Theorem	fnresdisj 3592	A function restricted to a class disjoint with its domain is empty.
⊢ (F Fn A → ((A ∩ B) = ∅ ↔ (F ↾ B) = ∅))
Theorem	2elresin 3593	Membership in two functions restricted by each other's domain.
⊢ ((F Fn A ⋀ G Fn B) → ((⟨x, y⟩ ∈ F ⋀ ⟨x, z⟩ ∈ G) ↔ (⟨x, y⟩ ∈ (F ↾ (A ∩ B)) ⋀ ⟨x, z⟩ ∈ (G ↾ (A ∩ B)))))
Theorem	fnssresb 3594	Restriction of a function with a subclass of its domain.
⊢ (F Fn A → ((F ↾ B) Fn B ↔ B ⊆ A))
Theorem	fnssres 3595	Restriction of a function with a subclass of its domain.
⊢ ((F Fn A ⋀ B ⊆ A) → (F ↾ B) Fn B)
Theorem	fnresin1 3596	Restriction of a function's domain with an intersection.
⊢ (F Fn A → (F ↾ (A ∩ B)) Fn (A ∩ B))
Theorem	fnresin2 3597	Restriction of a function's domain with an intersection.
⊢ (F Fn A → (F ↾ (B ∩ A)) Fn (B ∩ A))
Theorem	fnresi 3598	Functionality and domain of restricted identity.
⊢ (I ↾ A) Fn A
Theorem	fnima 3599	The image of a function's domain is its range.
⊢ (F Fn A → (F “ A) = ran F)
Theorem	fn0 3600	A function with empty domain is empty.
⊢ (F Fn ∅ ↔ F = ∅)