Theorem tgval2 12845

Description: Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 12858) that (topGen‘𝐵) is indeed a topology (on ∪ 𝐵, see unitg 12856). See also tgval 12843 and tgval3 12852. (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 12843	. 2 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})
2		inss1 3347	. . . . . . . . 9 ⊢ (𝐵 ∩ 𝒫 𝑥) ⊆ 𝐵
3	2	unissi 3819	. . . . . . . 8 ⊢ ∪ (𝐵 ∩ 𝒫 𝑥) ⊆ ∪ 𝐵
4	3	sseli 3143	. . . . . . 7 ⊢ (𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥) → 𝑦 ∈ ∪ 𝐵)
5	4	pm4.71ri 390	. . . . . 6 ⊢ (𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ (𝑦 ∈ ∪ 𝐵 ∧ 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥)))
6	5	ralbii 2476	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ 𝐵 ∧ 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥)))
7		r19.26 2596	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ 𝐵 ∧ 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥)) ↔ (∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥)))
8	6, 7	bitri 183	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ (∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥)))
9		dfss3 3137	. . . 4 ⊢ (𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥))
10		dfss3 3137	. . . . 5 ⊢ (𝑥 ⊆ ∪ 𝐵 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ 𝐵)
11		elin 3310	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝐵 ∩ 𝒫 𝑥) ↔ (𝑧 ∈ 𝐵 ∧ 𝑧 ∈ 𝒫 𝑥))
12	11	anbi2i 454	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)) ↔ (𝑦 ∈ 𝑧 ∧ (𝑧 ∈ 𝐵 ∧ 𝑧 ∈ 𝒫 𝑥)))
13		an12 556	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑧 ∧ (𝑧 ∈ 𝐵 ∧ 𝑧 ∈ 𝒫 𝑥)) ↔ (𝑧 ∈ 𝐵 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥)))
14	12, 13	bitri 183	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)) ↔ (𝑧 ∈ 𝐵 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥)))
15	14	exbii 1598	. . . . . . . 8 ⊢ (∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)) ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥)))
16		eluni 3799	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)))
17		df-rex 2454	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥) ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥)))
18	15, 16, 17	3bitr4i 211	. . . . . . 7 ⊢ (𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥))
19		velpw 3573	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝒫 𝑥 ↔ 𝑧 ⊆ 𝑥)
20	19	anbi2i 454	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥) ↔ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
21	20	rexbii 2477	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥) ↔ ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
22	18, 21	bitr2i 184	. . . . . 6 ⊢ (∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥) ↔ 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥))
23	22	ralbii 2476	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥) ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥))
24	10, 23	anbi12i 457	. . . 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 211	. . 3 ⊢ (𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ (𝑥 ⊆ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)))
26	25	abbii 2286	. 2 ⊢ {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} = {𝑥 ∣ (𝑥 ⊆ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))}
27	1, 26	eqtrdi 2219	1 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ (𝑥 ⊆ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))})

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348  ∃wex 1485   ∈ wcel 2141  {cab 2156  ∀wral 2448  ∃wrex 2449   ∩ cin 3120   ⊆ wss 3121  𝒫 cpw 3566  ∪ cuni 3796  ‘cfv 5198  topGenctg 12594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-topgen 12600

This theorem is referenced by:  eltg2  12847