Theorem tgval2 20571

Description: Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 20584) that (topGen‘𝐵) is indeed a topology (on ∪ 𝐵; see unitg 20582). (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)

Assertion

Ref	Expression
tgval2	⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ (𝑥 ⊆ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))})

Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥,𝑉,𝑦,𝑧

Proof of Theorem tgval2

Step	Hyp	Ref	Expression
1		tgval 20570	. 2 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})
2		inss1 3795	. . . . . . . . 9 ⊢ (𝐵 ∩ 𝒫 𝑥) ⊆ 𝐵
3	2	unissi 4397	. . . . . . . 8 ⊢ ∪ (𝐵 ∩ 𝒫 𝑥) ⊆ ∪ 𝐵
4	3	sseli 3564	. . . . . . 7 ⊢ (𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥) → 𝑦 ∈ ∪ 𝐵)
5	4	pm4.71ri 663	. . . . . 6 ⊢ (𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ (𝑦 ∈ ∪ 𝐵 ∧ 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥)))
6	5	ralbii 2963	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ 𝐵 ∧ 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥)))
7		r19.26 3046	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ 𝐵 ∧ 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥)) ↔ (∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥)))
8	6, 7	bitri 263	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ (∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥)))
9		dfss3 3558	. . . 4 ⊢ (𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥))
10		dfss3 3558	. . . . 5 ⊢ (𝑥 ⊆ ∪ 𝐵 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ 𝐵)
11		elin 3758	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝐵 ∩ 𝒫 𝑥) ↔ (𝑧 ∈ 𝐵 ∧ 𝑧 ∈ 𝒫 𝑥))
12	11	anbi2i 726	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)) ↔ (𝑦 ∈ 𝑧 ∧ (𝑧 ∈ 𝐵 ∧ 𝑧 ∈ 𝒫 𝑥)))
13		an12 834	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑧 ∧ (𝑧 ∈ 𝐵 ∧ 𝑧 ∈ 𝒫 𝑥)) ↔ (𝑧 ∈ 𝐵 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥)))
14	12, 13	bitri 263	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)) ↔ (𝑧 ∈ 𝐵 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥)))
15	14	exbii 1764	. . . . . . . 8 ⊢ (∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)) ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥)))
16		eluni 4375	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)))
17		df-rex 2902	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥) ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥)))
18	15, 16, 17	3bitr4i 291	. . . . . . 7 ⊢ (𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥))
19		selpw 4115	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝒫 𝑥 ↔ 𝑧 ⊆ 𝑥)
20	19	anbi2i 726	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥) ↔ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
21	20	rexbii 3023	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥) ↔ ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
22	18, 21	bitr2i 264	. . . . . 6 ⊢ (∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥) ↔ 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥))
23	22	ralbii 2963	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥) ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥))
24	10, 23	anbi12i 729	. . . 4 ⊢ ((𝑥 ⊆ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)) ↔ (∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥)))
25	8, 9, 24	3bitr4i 291	. . 3 ⊢ (𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ (𝑥 ⊆ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)))
26	25	abbii 2726	. 2 ⊢ {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} = {𝑥 ∣ (𝑥 ⊆ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))}
27	1, 26	syl6eq 2660	1 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ (𝑥 ⊆ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))})

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  ‘cfv 5804  topGenctg 15921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-topgen 15927

This theorem is referenced by:  eltg2  20573