Theorem tgval2 22934

Description: Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 22947) that (topGen‘𝐵) is indeed a topology (on ∪ 𝐵, see unitg 22945). See also tgval 22933 and tgval3 22941. (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 22933	. 2 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})
2		inss1 4178	. . . . . . . . 9 ⊢ (𝐵 ∩ 𝒫 𝑥) ⊆ 𝐵
3	2	unissi 4860	. . . . . . . 8 ⊢ ∪ (𝐵 ∩ 𝒫 𝑥) ⊆ ∪ 𝐵
4	3	sseli 3918	. . . . . . 7 ⊢ (𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥) → 𝑦 ∈ ∪ 𝐵)
5	4	pm4.71ri 560	. . . . . 6 ⊢ (𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ (𝑦 ∈ ∪ 𝐵 ∧ 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥)))
6	5	ralbii 3084	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ 𝐵 ∧ 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥)))
7		r19.26 3098	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ 𝐵 ∧ 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥)) ↔ (∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥)))
8	6, 7	bitri 275	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ (∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥)))
9		dfss3 3911	. . . 4 ⊢ (𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥))
10		dfss3 3911	. . . . 5 ⊢ (𝑥 ⊆ ∪ 𝐵 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ 𝐵)
11		elin 3906	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝐵 ∩ 𝒫 𝑥) ↔ (𝑧 ∈ 𝐵 ∧ 𝑧 ∈ 𝒫 𝑥))
12	11	anbi2i 624	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)) ↔ (𝑦 ∈ 𝑧 ∧ (𝑧 ∈ 𝐵 ∧ 𝑧 ∈ 𝒫 𝑥)))
13		an12 646	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑧 ∧ (𝑧 ∈ 𝐵 ∧ 𝑧 ∈ 𝒫 𝑥)) ↔ (𝑧 ∈ 𝐵 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥)))
14	12, 13	bitri 275	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)) ↔ (𝑧 ∈ 𝐵 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥)))
15	14	exbii 1850	. . . . . . . 8 ⊢ (∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)) ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥)))
16		eluni 4854	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)))
17		df-rex 3063	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥) ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥)))
18	15, 16, 17	3bitr4i 303	. . . . . . 7 ⊢ (𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥))
19		velpw 4547	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝒫 𝑥 ↔ 𝑧 ⊆ 𝑥)
20	19	anbi2i 624	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥) ↔ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
21	20	rexbii 3085	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥) ↔ ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
22	18, 21	bitr2i 276	. . . . . 6 ⊢ (∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥) ↔ 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥))
23	22	ralbii 3084	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥) ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥))
24	10, 23	anbi12i 629	. . . 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 303	. . 3 ⊢ (𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ (𝑥 ⊆ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)))
26	25	abbii 2804	. 2 ⊢ {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} = {𝑥 ∣ (𝑥 ⊆ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))}
27	1, 26	eqtrdi 2788	1 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ (𝑥 ⊆ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3062   ∩ cin 3889   ⊆ wss 3890  𝒫 cpw 4542  ∪ cuni 4851  ‘cfv 6493  topGenctg 17394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6449  df-fun 6495  df-fv 6501  df-topgen 17400

This theorem is referenced by:  eltg2  22936