qtopbaslem - Metamath Proof Explorer

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Theorem qtopbaslem 23922

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 13102	. . 3 ⊢ (,) ∈ V
2	1	imaex 7763	. 2 ⊢ ((,) “ (𝑆 × 𝑆)) ∈ V
3		qtopbas.1	. . . . . . . . 9 ⊢ 𝑆 ⊆ ℝ^*
4	3	sseli 3917	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ℝ^*)
5	3	sseli 3917	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑆 → 𝑤 ∈ ℝ^*)
6	4, 5	anim12i 613	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → (𝑧 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*))
7	3	sseli 3917	. . . . . . . 8 ⊢ (𝑣 ∈ 𝑆 → 𝑣 ∈ ℝ^*)
8	3	sseli 3917	. . . . . . . 8 ⊢ (𝑢 ∈ 𝑆 → 𝑢 ∈ ℝ^*)
9	7, 8	anim12i 613	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) → (𝑣 ∈ ℝ^* ∧ 𝑢 ∈ ℝ^*))
10		iooin 13113	. . . . . . 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 596	. . . . . 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 4504	. . . . . . . . 9 ⊢ ((𝑣 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → if(𝑧 ≤ 𝑣, 𝑣, 𝑧) ∈ 𝑆)
13	12	ancoms 459	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → if(𝑧 ≤ 𝑣, 𝑣, 𝑧) ∈ 𝑆)
14		ifcl 4504	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) → if(𝑤 ≤ 𝑢, 𝑤, 𝑢) ∈ 𝑆)
15		df-ov 7278	. . . . . . . . 9 ⊢ (if(𝑧 ≤ 𝑣, 𝑣, 𝑧)(,)if(𝑤 ≤ 𝑢, 𝑤, 𝑢)) = ((,)‘⟨if(𝑧 ≤ 𝑣, 𝑣, 𝑧), if(𝑤 ≤ 𝑢, 𝑤, 𝑢)⟩)
16		opelxpi 5626	. . . . . . . . . 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 13179	. . . . . . . . . . . 12 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
18		ffun 6603	. . . . . . . . . . . 12 ⊢ ((,):(ℝ^* × ℝ^*)⟶𝒫 ℝ → Fun (,))
19	17, 18	ax-mp 5	. . . . . . . . . . 11 ⊢ Fun (,)
20		xpss12 5604	. . . . . . . . . . . . 13 ⊢ ((𝑆 ⊆ ℝ^* ∧ 𝑆 ⊆ ℝ^) → (𝑆 × 𝑆) ⊆ (ℝ^ × ℝ^*))
21	3, 3, 20	mp2an 689	. . . . . . . . . . . 12 ⊢ (𝑆 × 𝑆) ⊆ (ℝ^* × ℝ^*)
22	17	fdmi 6612	. . . . . . . . . . . 12 ⊢ dom (,) = (ℝ^* × ℝ^*)
23	21, 22	sseqtrri 3958	. . . . . . . . . . 11 ⊢ (𝑆 × 𝑆) ⊆ dom (,)
24		funfvima2 7107	. . . . . . . . . . 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 689	. . . . . . . . . 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 2843	. . . . . . . 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 596	. . . . . . 7 ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝑤 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → (if(𝑧 ≤ 𝑣, 𝑣, 𝑧)(,)if(𝑤 ≤ 𝑢, 𝑤, 𝑢)) ∈ ((,) “ (𝑆 × 𝑆)))
29	28	an4s 657	. . . . . 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 2839	. . . . 5 ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆)))
31	30	ralrimivva 3123	. . . 4 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → ∀𝑣 ∈ 𝑆 ∀𝑢 ∈ 𝑆 ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆)))
32	31	rgen2 3120	. . 3 ⊢ ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 ∀𝑣 ∈ 𝑆 ∀𝑢 ∈ 𝑆 ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆))
33		ffn 6600	. . . . . 6 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
34	17, 33	ax-mp 5	. . . . 5 ⊢ (,) Fn (ℝ^* × ℝ^*)
35		ineq1 4139	. . . . . . . 8 ⊢ (𝑥 = ((,)‘𝑡) → (𝑥 ∩ 𝑦) = (((,)‘𝑡) ∩ 𝑦))
36	35	eleq1d 2823	. . . . . . 7 ⊢ (𝑥 = ((,)‘𝑡) → ((𝑥 ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ (((,)‘𝑡) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆))))
37	36	ralbidv 3112	. . . . . 6 ⊢ (𝑥 = ((,)‘𝑡) → (∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(𝑥 ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(((,)‘𝑡) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆))))
38	37	ralima 7114	. . . . 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 689	. . . 4 ⊢ (∀𝑥 ∈ ((,) “ (𝑆 × 𝑆))∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(𝑥 ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑡 ∈ (𝑆 × 𝑆)∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(((,)‘𝑡) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)))
40		fveq2 6774	. . . . . . . . . 10 ⊢ (𝑡 = ⟨𝑧, 𝑤⟩ → ((,)‘𝑡) = ((,)‘⟨𝑧, 𝑤⟩))
41		df-ov 7278	. . . . . . . . . 10 ⊢ (𝑧(,)𝑤) = ((,)‘⟨𝑧, 𝑤⟩)
42	40, 41	eqtr4di 2796	. . . . . . . . 9 ⊢ (𝑡 = ⟨𝑧, 𝑤⟩ → ((,)‘𝑡) = (𝑧(,)𝑤))
43	42	ineq1d 4145	. . . . . . . 8 ⊢ (𝑡 = ⟨𝑧, 𝑤⟩ → (((,)‘𝑡) ∩ 𝑦) = ((𝑧(,)𝑤) ∩ 𝑦))
44	43	eleq1d 2823	. . . . . . 7 ⊢ (𝑡 = ⟨𝑧, 𝑤⟩ → ((((,)‘𝑡) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ((𝑧(,)𝑤) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆))))
45	44	ralbidv 3112	. . . . . 6 ⊢ (𝑡 = ⟨𝑧, 𝑤⟩ → (∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(((,)‘𝑡) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))((𝑧(,)𝑤) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆))))
46		ineq2 4140	. . . . . . . . . 10 ⊢ (𝑦 = ((,)‘𝑡) → ((𝑧(,)𝑤) ∩ 𝑦) = ((𝑧(,)𝑤) ∩ ((,)‘𝑡)))
47	46	eleq1d 2823	. . . . . . . . 9 ⊢ (𝑦 = ((,)‘𝑡) → (((𝑧(,)𝑤) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ((𝑧(,)𝑤) ∩ ((,)‘𝑡)) ∈ ((,) “ (𝑆 × 𝑆))))
48	47	ralima 7114	. . . . . . . 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 689	. . . . . . 7 ⊢ (∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))((𝑧(,)𝑤) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑡 ∈ (𝑆 × 𝑆)((𝑧(,)𝑤) ∩ ((,)‘𝑡)) ∈ ((,) “ (𝑆 × 𝑆)))
50		fveq2 6774	. . . . . . . . . . 11 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → ((,)‘𝑡) = ((,)‘⟨𝑣, 𝑢⟩))
51		df-ov 7278	. . . . . . . . . . 11 ⊢ (𝑣(,)𝑢) = ((,)‘⟨𝑣, 𝑢⟩)
52	50, 51	eqtr4di 2796	. . . . . . . . . 10 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → ((,)‘𝑡) = (𝑣(,)𝑢))
53	52	ineq2d 4146	. . . . . . . . 9 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → ((𝑧(,)𝑤) ∩ ((,)‘𝑡)) = ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)))
54	53	eleq1d 2823	. . . . . . . 8 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → (((𝑧(,)𝑤) ∩ ((,)‘𝑡)) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆))))
55	54	ralxp 5750	. . . . . . 7 ⊢ (∀𝑡 ∈ (𝑆 × 𝑆)((𝑧(,)𝑤) ∩ ((,)‘𝑡)) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑣 ∈ 𝑆 ∀𝑢 ∈ 𝑆 ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆)))
56	49, 55	bitri 274	. . . . . 6 ⊢ (∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))((𝑧(,)𝑤) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑣 ∈ 𝑆 ∀𝑢 ∈ 𝑆 ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆)))
57	45, 56	bitrdi 287	. . . . 5 ⊢ (𝑡 = ⟨𝑧, 𝑤⟩ → (∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(((,)‘𝑡) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑣 ∈ 𝑆 ∀𝑢 ∈ 𝑆 ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆))))
58	57	ralxp 5750	. . . 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 274	. . 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 230	. 2 ⊢ ∀𝑥 ∈ ((,) “ (𝑆 × 𝑆))∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(𝑥 ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆))
61		fiinbas 22102	. 2 ⊢ ((((,) “ (𝑆 × 𝑆)) ∈ V ∧ ∀𝑥 ∈ ((,) “ (𝑆 × 𝑆))∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(𝑥 ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆))) → ((,) “ (𝑆 × 𝑆)) ∈ TopBases)
62	2, 60, 61	mp2an 689	1 ⊢ ((,) “ (𝑆 × 𝑆)) ∈ TopBases

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 Vcvv 3432 ∩ cin 3886 ⊆ wss 3887 ifcif 4459 𝒫 cpw 4533 ⟨cop 4567 class class class wbr 5074 × cxp 5587 dom cdm 5589 “ cima 5592 Fun wfun 6427 Fn wfn 6428 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ℝcr 10870 ℝ^*cxr 11008 ≤ cle 11010 (,)cioo 13079 TopBasesctb 22095

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-pre-lttri 10945 ax-pre-lttrn 10946

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-ioo 13083 df-bases 22096

This theorem is referenced by: qtopbas 23923 retopbas 23924

W3C validator