qtopbaslem - Metamath Proof Explorer

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Theorem qtopbaslem 23361

Description: The set of open intervals with endpoints in a subset forms a basis for a topology. (Contributed by Mario Carneiro, 17-Jun-2014.)

**Hypothesis**
Ref	Expression
qtopbas.1	⊢ 𝑆 ⊆ ℝ^*

**Assertion**
Ref	Expression
qtopbaslem	⊢ ((,) “ (𝑆 × 𝑆)) ∈ TopBases

Proof of Theorem qtopbaslem Dummy variables 𝑢 𝑡 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iooex 12755	. . 3 ⊢ (,) ∈ V
2	1	imaex 7615	. 2 ⊢ ((,) “ (𝑆 × 𝑆)) ∈ V
3		qtopbas.1	. . . . . . . . 9 ⊢ 𝑆 ⊆ ℝ^*
4	3	sseli 3962	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ℝ^*)
5	3	sseli 3962	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑆 → 𝑤 ∈ ℝ^*)
6	4, 5	anim12i 614	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → (𝑧 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*))
7	3	sseli 3962	. . . . . . . 8 ⊢ (𝑣 ∈ 𝑆 → 𝑣 ∈ ℝ^*)
8	3	sseli 3962	. . . . . . . 8 ⊢ (𝑢 ∈ 𝑆 → 𝑢 ∈ ℝ^*)
9	7, 8	anim12i 614	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) → (𝑣 ∈ ℝ^* ∧ 𝑢 ∈ ℝ^*))
10		iooin 12766	. . . . . . 7 ⊢ (((𝑧 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^) ∧ (𝑣 ∈ ℝ^ ∧ 𝑢 ∈ ℝ^*)) → ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) = (if(𝑧 ≤ 𝑣, 𝑣, 𝑧)(,)if(𝑤 ≤ 𝑢, 𝑤, 𝑢)))
11	6, 9, 10	syl2an 597	. . . . . 6 ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) = (if(𝑧 ≤ 𝑣, 𝑣, 𝑧)(,)if(𝑤 ≤ 𝑢, 𝑤, 𝑢)))
12		ifcl 4510	. . . . . . . . 9 ⊢ ((𝑣 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → if(𝑧 ≤ 𝑣, 𝑣, 𝑧) ∈ 𝑆)
13	12	ancoms 461	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → if(𝑧 ≤ 𝑣, 𝑣, 𝑧) ∈ 𝑆)
14		ifcl 4510	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) → if(𝑤 ≤ 𝑢, 𝑤, 𝑢) ∈ 𝑆)
15		df-ov 7153	. . . . . . . . 9 ⊢ (if(𝑧 ≤ 𝑣, 𝑣, 𝑧)(,)if(𝑤 ≤ 𝑢, 𝑤, 𝑢)) = ((,)‘⟨if(𝑧 ≤ 𝑣, 𝑣, 𝑧), if(𝑤 ≤ 𝑢, 𝑤, 𝑢)⟩)
16		opelxpi 5586	. . . . . . . . . 10 ⊢ ((if(𝑧 ≤ 𝑣, 𝑣, 𝑧) ∈ 𝑆 ∧ if(𝑤 ≤ 𝑢, 𝑤, 𝑢) ∈ 𝑆) → ⟨if(𝑧 ≤ 𝑣, 𝑣, 𝑧), if(𝑤 ≤ 𝑢, 𝑤, 𝑢)⟩ ∈ (𝑆 × 𝑆))
17		ioof 12829	. . . . . . . . . . . 12 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
18		ffun 6511	. . . . . . . . . . . 12 ⊢ ((,):(ℝ^* × ℝ^*)⟶𝒫 ℝ → Fun (,))
19	17, 18	ax-mp 5	. . . . . . . . . . 11 ⊢ Fun (,)
20		xpss12 5564	. . . . . . . . . . . . 13 ⊢ ((𝑆 ⊆ ℝ^* ∧ 𝑆 ⊆ ℝ^) → (𝑆 × 𝑆) ⊆ (ℝ^ × ℝ^*))
21	3, 3, 20	mp2an 690	. . . . . . . . . . . 12 ⊢ (𝑆 × 𝑆) ⊆ (ℝ^* × ℝ^*)
22	17	fdmi 6518	. . . . . . . . . . . 12 ⊢ dom (,) = (ℝ^* × ℝ^*)
23	21, 22	sseqtrri 4003	. . . . . . . . . . 11 ⊢ (𝑆 × 𝑆) ⊆ dom (,)
24		funfvima2 6987	. . . . . . . . . . 11 ⊢ ((Fun (,) ∧ (𝑆 × 𝑆) ⊆ dom (,)) → (⟨if(𝑧 ≤ 𝑣, 𝑣, 𝑧), if(𝑤 ≤ 𝑢, 𝑤, 𝑢)⟩ ∈ (𝑆 × 𝑆) → ((,)‘⟨if(𝑧 ≤ 𝑣, 𝑣, 𝑧), if(𝑤 ≤ 𝑢, 𝑤, 𝑢)⟩) ∈ ((,) “ (𝑆 × 𝑆))))
25	19, 23, 24	mp2an 690	. . . . . . . . . 10 ⊢ (⟨if(𝑧 ≤ 𝑣, 𝑣, 𝑧), if(𝑤 ≤ 𝑢, 𝑤, 𝑢)⟩ ∈ (𝑆 × 𝑆) → ((,)‘⟨if(𝑧 ≤ 𝑣, 𝑣, 𝑧), if(𝑤 ≤ 𝑢, 𝑤, 𝑢)⟩) ∈ ((,) “ (𝑆 × 𝑆)))
26	16, 25	syl 17	. . . . . . . . 9 ⊢ ((if(𝑧 ≤ 𝑣, 𝑣, 𝑧) ∈ 𝑆 ∧ if(𝑤 ≤ 𝑢, 𝑤, 𝑢) ∈ 𝑆) → ((,)‘⟨if(𝑧 ≤ 𝑣, 𝑣, 𝑧), if(𝑤 ≤ 𝑢, 𝑤, 𝑢)⟩) ∈ ((,) “ (𝑆 × 𝑆)))
27	15, 26	eqeltrid 2917	. . . . . . . 8 ⊢ ((if(𝑧 ≤ 𝑣, 𝑣, 𝑧) ∈ 𝑆 ∧ if(𝑤 ≤ 𝑢, 𝑤, 𝑢) ∈ 𝑆) → (if(𝑧 ≤ 𝑣, 𝑣, 𝑧)(,)if(𝑤 ≤ 𝑢, 𝑤, 𝑢)) ∈ ((,) “ (𝑆 × 𝑆)))
28	13, 14, 27	syl2an 597	. . . . . . 7 ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝑤 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → (if(𝑧 ≤ 𝑣, 𝑣, 𝑧)(,)if(𝑤 ≤ 𝑢, 𝑤, 𝑢)) ∈ ((,) “ (𝑆 × 𝑆)))
29	28	an4s 658	. . . . . 6 ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → (if(𝑧 ≤ 𝑣, 𝑣, 𝑧)(,)if(𝑤 ≤ 𝑢, 𝑤, 𝑢)) ∈ ((,) “ (𝑆 × 𝑆)))
30	11, 29	eqeltrd 2913	. . . . 5 ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆)))
31	30	ralrimivva 3191	. . . 4 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → ∀𝑣 ∈ 𝑆 ∀𝑢 ∈ 𝑆 ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆)))
32	31	rgen2 3203	. . 3 ⊢ ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 ∀𝑣 ∈ 𝑆 ∀𝑢 ∈ 𝑆 ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆))
33		ffn 6508	. . . . . 6 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
34	17, 33	ax-mp 5	. . . . 5 ⊢ (,) Fn (ℝ^* × ℝ^*)
35		ineq1 4180	. . . . . . . 8 ⊢ (𝑥 = ((,)‘𝑡) → (𝑥 ∩ 𝑦) = (((,)‘𝑡) ∩ 𝑦))
36	35	eleq1d 2897	. . . . . . 7 ⊢ (𝑥 = ((,)‘𝑡) → ((𝑥 ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ (((,)‘𝑡) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆))))
37	36	ralbidv 3197	. . . . . 6 ⊢ (𝑥 = ((,)‘𝑡) → (∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(𝑥 ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(((,)‘𝑡) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆))))
38	37	ralima 6994	. . . . 5 ⊢ (((,) Fn (ℝ^* × ℝ^) ∧ (𝑆 × 𝑆) ⊆ (ℝ^ × ℝ^*)) → (∀𝑥 ∈ ((,) “ (𝑆 × 𝑆))∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(𝑥 ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑡 ∈ (𝑆 × 𝑆)∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(((,)‘𝑡) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆))))
39	34, 21, 38	mp2an 690	. . . 4 ⊢ (∀𝑥 ∈ ((,) “ (𝑆 × 𝑆))∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(𝑥 ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑡 ∈ (𝑆 × 𝑆)∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(((,)‘𝑡) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)))
40		fveq2 6664	. . . . . . . . . 10 ⊢ (𝑡 = ⟨𝑧, 𝑤⟩ → ((,)‘𝑡) = ((,)‘⟨𝑧, 𝑤⟩))
41		df-ov 7153	. . . . . . . . . 10 ⊢ (𝑧(,)𝑤) = ((,)‘⟨𝑧, 𝑤⟩)
42	40, 41	syl6eqr 2874	. . . . . . . . 9 ⊢ (𝑡 = ⟨𝑧, 𝑤⟩ → ((,)‘𝑡) = (𝑧(,)𝑤))
43	42	ineq1d 4187	. . . . . . . 8 ⊢ (𝑡 = ⟨𝑧, 𝑤⟩ → (((,)‘𝑡) ∩ 𝑦) = ((𝑧(,)𝑤) ∩ 𝑦))
44	43	eleq1d 2897	. . . . . . 7 ⊢ (𝑡 = ⟨𝑧, 𝑤⟩ → ((((,)‘𝑡) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ((𝑧(,)𝑤) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆))))
45	44	ralbidv 3197	. . . . . 6 ⊢ (𝑡 = ⟨𝑧, 𝑤⟩ → (∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(((,)‘𝑡) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))((𝑧(,)𝑤) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆))))
46		ineq2 4182	. . . . . . . . . 10 ⊢ (𝑦 = ((,)‘𝑡) → ((𝑧(,)𝑤) ∩ 𝑦) = ((𝑧(,)𝑤) ∩ ((,)‘𝑡)))
47	46	eleq1d 2897	. . . . . . . . 9 ⊢ (𝑦 = ((,)‘𝑡) → (((𝑧(,)𝑤) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ((𝑧(,)𝑤) ∩ ((,)‘𝑡)) ∈ ((,) “ (𝑆 × 𝑆))))
48	47	ralima 6994	. . . . . . . 8 ⊢ (((,) Fn (ℝ^* × ℝ^) ∧ (𝑆 × 𝑆) ⊆ (ℝ^ × ℝ^*)) → (∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))((𝑧(,)𝑤) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑡 ∈ (𝑆 × 𝑆)((𝑧(,)𝑤) ∩ ((,)‘𝑡)) ∈ ((,) “ (𝑆 × 𝑆))))
49	34, 21, 48	mp2an 690	. . . . . . 7 ⊢ (∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))((𝑧(,)𝑤) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑡 ∈ (𝑆 × 𝑆)((𝑧(,)𝑤) ∩ ((,)‘𝑡)) ∈ ((,) “ (𝑆 × 𝑆)))
50		fveq2 6664	. . . . . . . . . . 11 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → ((,)‘𝑡) = ((,)‘⟨𝑣, 𝑢⟩))
51		df-ov 7153	. . . . . . . . . . 11 ⊢ (𝑣(,)𝑢) = ((,)‘⟨𝑣, 𝑢⟩)
52	50, 51	syl6eqr 2874	. . . . . . . . . 10 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → ((,)‘𝑡) = (𝑣(,)𝑢))
53	52	ineq2d 4188	. . . . . . . . 9 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → ((𝑧(,)𝑤) ∩ ((,)‘𝑡)) = ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)))
54	53	eleq1d 2897	. . . . . . . 8 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → (((𝑧(,)𝑤) ∩ ((,)‘𝑡)) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆))))
55	54	ralxp 5706	. . . . . . 7 ⊢ (∀𝑡 ∈ (𝑆 × 𝑆)((𝑧(,)𝑤) ∩ ((,)‘𝑡)) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑣 ∈ 𝑆 ∀𝑢 ∈ 𝑆 ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆)))
56	49, 55	bitri 277	. . . . . 6 ⊢ (∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))((𝑧(,)𝑤) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑣 ∈ 𝑆 ∀𝑢 ∈ 𝑆 ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆)))
57	45, 56	syl6bb 289	. . . . 5 ⊢ (𝑡 = ⟨𝑧, 𝑤⟩ → (∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(((,)‘𝑡) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑣 ∈ 𝑆 ∀𝑢 ∈ 𝑆 ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆))))
58	57	ralxp 5706	. . . 4 ⊢ (∀𝑡 ∈ (𝑆 × 𝑆)∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(((,)‘𝑡) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 ∀𝑣 ∈ 𝑆 ∀𝑢 ∈ 𝑆 ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆)))
59	39, 58	bitri 277	. . 3 ⊢ (∀𝑥 ∈ ((,) “ (𝑆 × 𝑆))∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(𝑥 ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 ∀𝑣 ∈ 𝑆 ∀𝑢 ∈ 𝑆 ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆)))
60	32, 59	mpbir 233	. 2 ⊢ ∀𝑥 ∈ ((,) “ (𝑆 × 𝑆))∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(𝑥 ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆))
61		fiinbas 21554	. 2 ⊢ ((((,) “ (𝑆 × 𝑆)) ∈ V ∧ ∀𝑥 ∈ ((,) “ (𝑆 × 𝑆))∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(𝑥 ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆))) → ((,) “ (𝑆 × 𝑆)) ∈ TopBases)
62	2, 60, 61	mp2an 690	1 ⊢ ((,) “ (𝑆 × 𝑆)) ∈ TopBases

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 ∩ cin 3934 ⊆ wss 3935 ifcif 4466 𝒫 cpw 4538 ⟨cop 4566 class class class wbr 5058 × cxp 5547 dom cdm 5549 “ cima 5552 Fun wfun 6343 Fn wfn 6344 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ℝcr 10530 ℝ^*cxr 10668 ≤ cle 10670 (,)cioo 12732 TopBasesctb 21547

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-pre-lttri 10605 ax-pre-lttrn 10606

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-ioo 12736 df-bases 21548

This theorem is referenced by: qtopbas 23362 retopbas 23363

W3C validator