qtopbaslem - Metamath Proof Explorer

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Theorem qtopbaslem 24748

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 13319	. . 3 ⊢ (,) ∈ V
2	1	imaex 7861	. 2 ⊢ ((,) “ (𝑆 × 𝑆)) ∈ V
3		qtopbas.1	. . . . . . . . 9 ⊢ 𝑆 ⊆ ℝ^*
4	3	sseli 3918	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ℝ^*)
5	3	sseli 3918	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑆 → 𝑤 ∈ ℝ^*)
6	4, 5	anim12i 619	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → (𝑧 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*))
7	3	sseli 3918	. . . . . . . 8 ⊢ (𝑣 ∈ 𝑆 → 𝑣 ∈ ℝ^*)
8	3	sseli 3918	. . . . . . . 8 ⊢ (𝑢 ∈ 𝑆 → 𝑢 ∈ ℝ^*)
9	7, 8	anim12i 619	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) → (𝑣 ∈ ℝ^* ∧ 𝑢 ∈ ℝ^*))
10		iooin 13330	. . . . . . 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 602	. . . . . 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 4507	. . . . . . . . 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 4507	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) → if(𝑤 ≤ 𝑢, 𝑤, 𝑢) ∈ 𝑆)
15		df-ov 7366	. . . . . . . . 9 ⊢ (if(𝑧 ≤ 𝑣, 𝑣, 𝑧)(,)if(𝑤 ≤ 𝑢, 𝑤, 𝑢)) = ((,)‘⟨if(𝑧 ≤ 𝑣, 𝑣, 𝑧), if(𝑤 ≤ 𝑢, 𝑤, 𝑢)⟩)
16		opelxpi 5662	. . . . . . . . . 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 13398	. . . . . . . . . . . 12 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
18		ffun 6665	. . . . . . . . . . . 12 ⊢ ((,):(ℝ^* × ℝ^*)⟶𝒫 ℝ → Fun (,))
19	17, 18	ax-mp 5	. . . . . . . . . . 11 ⊢ Fun (,)
20		xpss12 5640	. . . . . . . . . . . . 13 ⊢ ((𝑆 ⊆ ℝ^* ∧ 𝑆 ⊆ ℝ^) → (𝑆 × 𝑆) ⊆ (ℝ^ × ℝ^*))
21	3, 3, 20	mp2an 698	. . . . . . . . . . . 12 ⊢ (𝑆 × 𝑆) ⊆ (ℝ^* × ℝ^*)
22	17	fdmi 6673	. . . . . . . . . . . 12 ⊢ dom (,) = (ℝ^* × ℝ^*)
23	21, 22	sseqtrri 3971	. . . . . . . . . . 11 ⊢ (𝑆 × 𝑆) ⊆ dom (,)
24		funfvima2 7182	. . . . . . . . . . 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 698	. . . . . . . . . 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 2844	. . . . . . . 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 602	. . . . . . 7 ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝑤 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → (if(𝑧 ≤ 𝑣, 𝑣, 𝑧)(,)if(𝑤 ≤ 𝑢, 𝑤, 𝑢)) ∈ ((,) “ (𝑆 × 𝑆)))
29	28	an4s 666	. . . . . 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 2840	. . . . 5 ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆)))
31	30	ralrimivva 3183	. . . 4 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → ∀𝑣 ∈ 𝑆 ∀𝑢 ∈ 𝑆 ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆)))
32	31	rgen2 3180	. . 3 ⊢ ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 ∀𝑣 ∈ 𝑆 ∀𝑢 ∈ 𝑆 ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆))
33		ffn 6662	. . . . . 6 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
34	17, 33	ax-mp 5	. . . . 5 ⊢ (,) Fn (ℝ^* × ℝ^*)
35		ineq1 4149	. . . . . . . 8 ⊢ (𝑥 = ((,)‘𝑡) → (𝑥 ∩ 𝑦) = (((,)‘𝑡) ∩ 𝑦))
36	35	eleq1d 2825	. . . . . . 7 ⊢ (𝑥 = ((,)‘𝑡) → ((𝑥 ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ (((,)‘𝑡) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆))))
37	36	ralbidv 3163	. . . . . 6 ⊢ (𝑥 = ((,)‘𝑡) → (∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(𝑥 ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(((,)‘𝑡) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆))))
38	37	ralima 7188	. . . . 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 698	. . . 4 ⊢ (∀𝑥 ∈ ((,) “ (𝑆 × 𝑆))∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(𝑥 ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑡 ∈ (𝑆 × 𝑆)∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(((,)‘𝑡) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)))
40		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑡 = ⟨𝑧, 𝑤⟩ → ((,)‘𝑡) = ((,)‘⟨𝑧, 𝑤⟩))
41		df-ov 7366	. . . . . . . . . 10 ⊢ (𝑧(,)𝑤) = ((,)‘⟨𝑧, 𝑤⟩)
42	40, 41	eqtr4di 2793	. . . . . . . . 9 ⊢ (𝑡 = ⟨𝑧, 𝑤⟩ → ((,)‘𝑡) = (𝑧(,)𝑤))
43	42	ineq1d 4155	. . . . . . . 8 ⊢ (𝑡 = ⟨𝑧, 𝑤⟩ → (((,)‘𝑡) ∩ 𝑦) = ((𝑧(,)𝑤) ∩ 𝑦))
44	43	eleq1d 2825	. . . . . . 7 ⊢ (𝑡 = ⟨𝑧, 𝑤⟩ → ((((,)‘𝑡) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ((𝑧(,)𝑤) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆))))
45	44	ralbidv 3163	. . . . . 6 ⊢ (𝑡 = ⟨𝑧, 𝑤⟩ → (∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(((,)‘𝑡) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))((𝑧(,)𝑤) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆))))
46		ineq2 4150	. . . . . . . . . 10 ⊢ (𝑦 = ((,)‘𝑡) → ((𝑧(,)𝑤) ∩ 𝑦) = ((𝑧(,)𝑤) ∩ ((,)‘𝑡)))
47	46	eleq1d 2825	. . . . . . . . 9 ⊢ (𝑦 = ((,)‘𝑡) → (((𝑧(,)𝑤) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ((𝑧(,)𝑤) ∩ ((,)‘𝑡)) ∈ ((,) “ (𝑆 × 𝑆))))
48	47	ralima 7188	. . . . . . . 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 698	. . . . . . 7 ⊢ (∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))((𝑧(,)𝑤) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑡 ∈ (𝑆 × 𝑆)((𝑧(,)𝑤) ∩ ((,)‘𝑡)) ∈ ((,) “ (𝑆 × 𝑆)))
50		fveq2 6834	. . . . . . . . . . 11 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → ((,)‘𝑡) = ((,)‘⟨𝑣, 𝑢⟩))
51		df-ov 7366	. . . . . . . . . . 11 ⊢ (𝑣(,)𝑢) = ((,)‘⟨𝑣, 𝑢⟩)
52	50, 51	eqtr4di 2793	. . . . . . . . . 10 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → ((,)‘𝑡) = (𝑣(,)𝑢))
53	52	ineq2d 4156	. . . . . . . . 9 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → ((𝑧(,)𝑤) ∩ ((,)‘𝑡)) = ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)))
54	53	eleq1d 2825	. . . . . . . 8 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → (((𝑧(,)𝑤) ∩ ((,)‘𝑡)) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆))))
55	54	ralxp 5790	. . . . . . 7 ⊢ (∀𝑡 ∈ (𝑆 × 𝑆)((𝑧(,)𝑤) ∩ ((,)‘𝑡)) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑣 ∈ 𝑆 ∀𝑢 ∈ 𝑆 ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆)))
56	49, 55	bitri 276	. . . . . 6 ⊢ (∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))((𝑧(,)𝑤) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑣 ∈ 𝑆 ∀𝑢 ∈ 𝑆 ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆)))
57	45, 56	bitrdi 288	. . . . 5 ⊢ (𝑡 = ⟨𝑧, 𝑤⟩ → (∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(((,)‘𝑡) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑣 ∈ 𝑆 ∀𝑢 ∈ 𝑆 ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆))))
58	57	ralxp 5790	. . . 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 276	. . 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 232	. 2 ⊢ ∀𝑥 ∈ ((,) “ (𝑆 × 𝑆))∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(𝑥 ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆))
61		fiinbas 22942	. 2 ⊢ ((((,) “ (𝑆 × 𝑆)) ∈ V ∧ ∀𝑥 ∈ ((,) “ (𝑆 × 𝑆))∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(𝑥 ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆))) → ((,) “ (𝑆 × 𝑆)) ∈ TopBases)
62	2, 60, 61	mp2an 698	1 ⊢ ((,) “ (𝑆 × 𝑆)) ∈ TopBases

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ∀wral 3054 Vcvv 3432 ∩ cin 3889 ⊆ wss 3890 ifcif 4461 𝒫 cpw 4536 ⟨cop 4568 class class class wbr 5079 × cxp 5623 dom cdm 5625 “ cima 5628 Fun wfun 6486 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 (class class class)co 7363 ℝcr 11035 ℝ^*cxr 11176 ≤ cle 11178 (,)cioo 13296 TopBasesctb 22935

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-pre-lttri 11110 ax-pre-lttrn 11111

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-id 5520 df-po 5533 df-so 5534 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7366 df-oprab 7367 df-mpo 7368 df-1st 7938 df-2nd 7939 df-er 8640 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-ioo 13300 df-bases 22936

This theorem is referenced by: qtopbas 24749 retopbas 24750

W3C validator