Metamath Proof Explorer - mmtheorems26

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Statement List for Metamath Proof Explorer - 2501-2600 - Page 26 of 58

Type	Label	Description
Statement
Theorem	rele 2501	The membership relation is a relation.
⊢ Rel E
Theorem	coeq1 2502	Equality theorem for composition of two classes.
⊢ (A = B → (A ∘ C) = (B ∘ C))
Theorem	coeq2 2503	Equality theorem for composition of two classes.
⊢ (A = B → (C ∘ A) = (C ∘ B))
Theorem	coeq1i 2504	Equality inference for composition of two classes.
⊢ A = B    ⇒   ⊢ (A ∘ C) = (B ∘ C)
Theorem	coeq2i 2505	Equality inference for composition of two classes.
⊢ A = B    ⇒   ⊢ (C ∘ A) = (C ∘ B)
Theorem	coeq1d 2506	Equality deduction for composition of two classes.
⊢ (φ → A = B)    ⇒   ⊢ (φ → (A ∘ C) = (B ∘ C))
Theorem	coeq2d 2507	Equality deduction for composition of two classes.
⊢ (φ → A = B)    ⇒   ⊢ (φ → (C ∘ A) = (C ∘ B))
Theorem	hbco 2508	Bound-variable hypothesis builder for function value.
⊢ (y ∈ A → ∀x y ∈ A)    &   ⊢ (y ∈ B → ∀x y ∈ B)    ⇒   ⊢ (y ∈ (A ∘ B) → ∀x y ∈ (A ∘ B))
Theorem	opelco 2509	Ordered pair membership in a composition.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ (⟨A, B⟩ ∈ (C ∘ D) ↔ ∃x(ADx ∧ xCB))
Theorem	brco 2510	Binary relation on a composition.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ (A(C ∘ D)B ↔ ∃x(ADx ∧ xCB))
Theorem	opelcog 2511	Ordered pair membership in a composition.
⊢ ((A ∈ R ∧ B ∈ S) → (⟨A, B⟩ ∈ (C ∘ D) ↔ ∃x(⟨A, x⟩ ∈ D ∧ ⟨x, B⟩ ∈ C)))
Theorem	cnvss 2512	Subset theorem for converse.
⊢ (A ⊆ B → ◡A ⊆ ◡B)
Theorem	cnveq 2513	Equality theorem for converse.
⊢ (A = B → ◡A = ◡B)
Theorem	elcnv 2514	Membership in a converse. Equation (5) of [Suppes] p. 62.
⊢ (A ∈ ◡R ↔ ∃x∃y(A = ⟨x, y⟩ ∧ yRx))
Theorem	elcnv2 2515	Membership in a converse. Equation (5) of [Suppes] p. 62.
⊢ (A ∈ ◡R ↔ ∃x∃y(A = ⟨x, y⟩ ∧ ⟨y, x⟩ ∈ R))
Theorem	hbcnv 2516	Bound-variable hypothesis builder for converse.
⊢ (y ∈ A → ∀x y ∈ A)    ⇒   ⊢ (y ∈ ◡A → ∀x y ∈ ◡A)
Theorem	opelcnvg 2517	Ordered-pair membership in converse.
⊢ ((A ∈ C ∧ B ∈ D) → (⟨A, B⟩ ∈ ◡R ↔ ⟨B, A⟩ ∈ R))
Theorem	opelcnv 2518	Ordered-pair membership in converse.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ (⟨A, B⟩ ∈ ◡R ↔ ⟨B, A⟩ ∈ R)
Theorem	brcnv 2519	The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ (A◡RB ↔ BRA)
Theorem	cnvco 2520	Distributive law of converse over class composition. Theorem 26 of [Suppes] p. 64.
⊢ ◡(A ∘ B) = (◡B ∘ ◡A)
Theorem	cnvuni 2521	The converse of a class union is the (indexed) union of the converses of its members.
⊢ ◡∪A = ∪x ∈ A ◡x
Theorem	dfdm3 2522	Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24.
⊢ dom A = {x∣∃y⟨x, y⟩ ∈ A}
Theorem	dfrn2 2523	Alternate definition of range. Definition 4 of [Suppes] p. 60.
⊢ ran A = {y∣∃x xAy}
Theorem	dfrn3 2524	Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24.
⊢ ran A = {y∣∃x⟨x, y⟩ ∈ A}
Theorem	dfdm4 2525	Alternate definition of domain.
⊢ dom A = ran ◡A
Theorem	dfdmf 2526	Domain definition requiring only that x and y not be 'free' in A (but not necessarily absent from it).
⊢ (z ∈ A → ∀x z ∈ A)    &   ⊢ (z ∈ A → ∀y z ∈ A)    ⇒   ⊢ dom A = {x∣∃y⟨x, y⟩ ∈ A}
Theorem	eldm 2527	Membership in a domain. Theorem 4 of [Suppes] p. 59.
⊢ A ∈ V    ⇒   ⊢ (A ∈ dom B ↔ ∃y ABy)
Theorem	eldm2 2528	Membership in a domain. Theorem 4 of [Suppes] p. 59.
⊢ A ∈ V    ⇒   ⊢ (A ∈ dom B ↔ ∃y⟨A, y⟩ ∈ B)
Theorem	eldmg 2529	Domain membership. Theorem 4 of [Suppes] p. 59.
⊢ (A ∈ C → (A ∈ dom B ↔ ∃y⟨A, y⟩ ∈ B))
Theorem	dmss 2530	Subset theorem for domain.
⊢ (A ⊆ B → dom A ⊆ dom B)
Theorem	dmeq 2531	Equality theorem for domain.
⊢ (A = B → dom A = dom B)
Theorem	dmeqi 2532	Equality inference for domain.
⊢ A = B    ⇒   ⊢ dom A = dom B
Theorem	dmeqd 2533	Equality deduction for domain.
⊢ (φ → A = B)    ⇒   ⊢ (φ → dom A = dom B)
Theorem	opeldm 2534	Membership of first of an ordered pair in a domain.
⊢ A ∈ V    ⇒   ⊢ (⟨A, B⟩ ∈ C → A ∈ dom C)
Theorem	breldm 2535	Membership of first of a binary relation in a domain.
⊢ A ∈ V    ⇒   ⊢ (ARB → A ∈ dom R)
Theorem	dmun 2536	The domain of a union is the union of domains. Exercise 56(a) of [Enderton] p. 65.
⊢ dom (A ∪ B) = (dom A ∪ dom B)
Theorem	dmin 2537	The domain of an intersection belong to the intersection of domains. Theorem 6 of [Suppes] p. 60.
⊢ dom (A ∩ B) ⊆ (dom A ∩ dom B)
Theorem	dmuni 2538	The domain of a union. Part of Exercise 8 of [Enderton] p. 41.
⊢ dom ∪A = ∪x ∈ A dom x
Theorem	dmopab 2539	The domain of a class of ordered pairs.
⊢ dom {⟨x, y⟩∣φ} = {x∣∃yφ}
Theorem	dmopabss 2540	Upper bound for the domain of a restricted class of ordered pairs.
⊢ dom {⟨x, y⟩∣(x ∈ A ∧ φ)} ⊆ A
Theorem	dmopab2 2541	The domain of a restricted class of ordered pairs.
⊢ (∀x ∈ A ∃yφ ↔ dom {⟨x, y⟩∣(x ∈ A ∧ φ)} = A)
Theorem	dm0 2542	The domain of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36.
⊢ dom ∅ = ∅
Theorem	dmsn0 2543	The domain of the singleton of the empty set is empty.
⊢ dom {∅} = ∅
Theorem	dmsnsn0 2544	The domain of the singleton of the singleton of the empty set is empty.
⊢ dom {{∅}} = ∅
Theorem	dmi 2545	The domain of the identity relation is the universe.
⊢ dom I = V
Theorem	dmv 2546	The domain of the universe is the universe.
⊢ dom V = V
Theorem	dmsnop 2547	The domain of a singleton of an ordered pair is the singleton of the first member.
⊢ dom {⟨A, B⟩} = {A}
Theorem	dmsnsnsn 2548	The domain of the singleton of the singleton of a singleton.
⊢ dom {{{A}}} = {A}
Theorem	dm0rn0 2549	An empty domain implies an empty range.
⊢ (dom A = ∅ ↔ ran A = ∅)
Theorem	reldm0 2550	A relation is empty iff its domain is empty.
⊢ (Rel A → (A = ∅ ↔ dom A = ∅))
Theorem	dmexg 2551	The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26.
⊢ (A ∈ B → dom A ∈ V)
Theorem	dmxp 2552	The domain of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37.
⊢ (¬ B = ∅ → dom (A × B) = A)
Theorem	dmxpid 2553	The domain of a cross product.
⊢ dom (A × A) = A
Theorem	elreldm 2554	The first member of an ordered pair in a relation belongs to the domain of the relation.
⊢ ((Rel A ∧ B ∈ A) → ∩∩B ∈ dom A)
Theorem	rneq 2555	Equality theorem for range.
⊢ (A = B → ran A = ran B)
Theorem	rneqi 2556	Equality inference for range.
⊢ A = B    ⇒   ⊢ ran A = ran B
Theorem	rneqd 2557	Equality deduction for range.
⊢ (φ → A = B)    ⇒   ⊢ (φ → ran A = ran B)
Theorem	rnss 2558	Subset theorem for range.
⊢ (A ⊆ B → ran A ⊆ ran B)
Theorem	brelrn 2559	The second argument of a binary relation belongs to its range.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ (ACB → B ∈ ran C)
Theorem	opelrn 2560	Membership of second member of an ordered pair in a range.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ (⟨A, B⟩ ∈ C → B ∈ ran C)
Theorem	dfrnf 2561	Range definition requiring only that x and y not be 'free' in A (but not necessarily absent from it).
⊢ (z ∈ A → ∀x z ∈ A)    &   ⊢ (z ∈ A → ∀y z ∈ A)    ⇒   ⊢ ran A = {y∣∃x⟨x, y⟩ ∈ A}
Theorem	elrn 2562	Membership in a range.
⊢ A ∈ V    ⇒   ⊢ (A ∈ ran B ↔ ∃x⟨x, A⟩ ∈ B)
Theorem	elrn2 2563	Membership in a range.
⊢ A ∈ V    ⇒   ⊢ (A ∈ ran B ↔ ∃x xBA)
Theorem	hbrn 2564	Bound-variable hypothesis builder for range.
⊢ (y ∈ A → ∀x y ∈ A)    ⇒   ⊢ (y ∈ ran A → ∀x y ∈ ran A)
Theorem	hbdm 2565	Bound-variable hypothesis builder for domain.
⊢ (y ∈ A → ∀x y ∈ A)    ⇒   ⊢ (y ∈ dom A → ∀x y ∈ dom A)
Theorem	rnopab 2566	The range of a class of ordered pairs.
⊢ ran {⟨x, y⟩∣φ} = {y∣∃xφ}
Theorem	rn0 2567	The range of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36.
⊢ ran ∅ = ∅
Theorem	relrn0 2568	A relation is empty iff its range is empty.
⊢ (Rel A → (A = ∅ ↔ ran A = ∅))
Theorem	rnexg 2569	The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26. Similar to Lemma 3D of [Enderton] p. 41.
⊢ (A ∈ B → ran A ∈ V)
Theorem	dmco 2570	Domain of a composition. Theorem 21 of [Suppes] p. 63.
⊢ dom (A ∘ B) ⊆ dom B
Theorem	rnco 2571	Range of a composition.
⊢ ran (A ∘ B) ⊆ ran A
Theorem	dmcosseq 2572	Domain of a composition.
⊢ (ran B ⊆ dom A → dom (A ∘ B) = dom B)
Theorem	dmcoeq 2573	Domain of a composition.
⊢ (dom A = ran B → dom (A ∘ B) = dom B)
Theorem	rncoeq 2574	Range of a composition.
⊢ (dom A = ran B → ran (A ∘ B) = ran A)
Theorem	reseq1 2575	Equality theorem for restrictions.
⊢ (A = B → (A ↾ C) = (B ↾ C))
Theorem	reseq2 2576	Equality theorem for restrictions.
⊢ (A = B → (C ↾ A) = (C ↾ B))
Theorem	hbres 2577	Bound-variable hypothesis builder for restriction.
⊢ (y ∈ A → ∀x y ∈ A)    &   ⊢ (y ∈ B → ∀x y ∈ B)    ⇒   ⊢ (y ∈ (A ↾ B) → ∀x y ∈ (A ↾ B))
Theorem	res0 2578	A restriction to the empty set is empty.
⊢ (A ↾ ∅) = ∅
Theorem	opelres 2579	Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25.
⊢ B ∈ V    ⇒   ⊢ (⟨A, B⟩ ∈ (C ↾ D) ↔ (⟨A, B⟩ ∈ C ∧ A ∈ D))
Theorem	opres 2580	Ordered pair membership in a restriction when the first member belongs to the restricting class.
⊢ B ∈ V    ⇒   ⊢ (A ∈ D → (⟨A, B⟩ ∈ (C ↾ D) ↔ ⟨A, B⟩ ∈ C))
Theorem	resieq 2581	A restricted identity relation is equivalent to equality in its domain.
⊢ ((B ∈ A ∧ C ∈ A) → (B(I ↾ A)C ↔ B = C))
Theorem	resundi 2582	Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65.
⊢ (A ↾ (B ∪ C)) = ((A ↾ B) ∪ (A ↾ C))
Theorem	resundir 2583	Distributive law for restriction over union.
⊢ ((A ∪ B) ↾ C) = ((A ↾ C) ∪ (B ↾ C))
Theorem	dmres 2584	The domain of a restriction. Exercise 14 of [TakeutiZaring] p. 25.
⊢ dom (A ↾ B) = (B ∩ dom A)
Theorem	ssdmres 2585	A domain restricted to a subclass equals the subclass.
⊢ (A ⊆ dom B ↔ dom (B ↾ A) = A)
Theorem	dmresexg 2586	The domain of a restriction to a set exists.
⊢ (B ∈ C → dom (A ↾ B) ∈ V)
Theorem	resss 2587	A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25.
⊢ (A ↾ B) ⊆ A
Theorem	rescom 2588	Commutative law for restriction.
⊢ ((A ↾ B) ↾ C) = ((A ↾ C) ↾ B)
Theorem	ssres 2589	Subclass theorem for restriction.
⊢ (A ⊆ B → (A ↾ C) ⊆ (B ↾ C))
Theorem	ssres2 2590	Subclass theorem for restriction.
⊢ (A ⊆ B → (C ↾ A) ⊆ (C ↾ B))
Theorem	relres 2591	A restriction is a relation. Exercise 12 of [TakeutiZaring] p. 25.
⊢ Rel (A ↾ B)
Theorem	resabs1 2592	Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25.
⊢ (B ⊆ C → ((A ↾ C) ↾ B) = (A ↾ B))
Theorem	resabs2 2593	Absorption law for restriction.
⊢ (B ⊆ C → ((A ↾ B) ↾ C) = (A ↾ B))
Theorem	residm 2594	Idempotent law for restriction.
⊢ ((A ↾ B) ↾ B) = (A ↾ B)
Theorem	resima 2595	A restriction to an image.
⊢ ((A ↾ B) “ B) = (A “ B)
Theorem	relssres 2596	Simplification law for restriction.
⊢ ((Rel A ∧ dom A ⊆ B) → (A ↾ B) = A)
Theorem	resexg 2597	The restriction of a set is a set.
⊢ (A ∈ C → (A ↾ B) ∈ V)
Theorem	resopab 2598	Restriction of a class abstraction of ordered pairs.
⊢ ({⟨x, y⟩∣φ} ↾ A) = {⟨x, y⟩∣(x ∈ A ∧ φ)}
Theorem	iss 2599	A subclass of the identity function is the identity function restricted to its domain.
⊢ (A ⊆ I ↔ A = (I ↾ dom A))
Theorem	dmresi 2600	The domain of the restricted identity function.
⊢ dom (I ↾ A) = A