Theorem eltg 22870

Description: Membership in a topology generated by a basis. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)

Assertion

Ref	Expression
eltg	⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴)))

Proof of Theorem eltg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tgval 22868	. . 3 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})
2	1	eleq2d 2817	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 ∈ {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)}))
3		elex 3457	. . . 4 ⊢ (𝐴 ∈ {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} → 𝐴 ∈ V)
4	3	adantl 481	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)}) → 𝐴 ∈ V)
5		inex1g 5257	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∩ 𝒫 𝐴) ∈ V)
6	5	uniexd 7675	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → ∪ (𝐵 ∩ 𝒫 𝐴) ∈ V)
7		ssexg 5261	. . . . 5 ⊢ ((𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴) ∧ ∪ (𝐵 ∩ 𝒫 𝐴) ∈ V) → 𝐴 ∈ V)
8	6, 7	sylan2 593	. . . 4 ⊢ ((𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴) ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V)
9	8	ancoms 458	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴)) → 𝐴 ∈ V)
10		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
11		pweq 4564	. . . . . . 7 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
12	11	ineq2d 4170	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝐴))
13	12	unieqd 4872	. . . . 5 ⊢ (𝑥 = 𝐴 → ∪ (𝐵 ∩ 𝒫 𝑥) = ∪ (𝐵 ∩ 𝒫 𝐴))
14	10, 13	sseq12d 3968	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴)))
15	14	elabg 3632	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} ↔ 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴)))
16	4, 9, 15	pm5.21nd 801	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} ↔ 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴)))
17	2, 16	bitrd 279	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2111  {cab 2709  Vcvv 3436   ∩ cin 3901   ⊆ wss 3902  𝒫 cpw 4550  ∪ cuni 4859  ‘cfv 6481  topGenctg 17338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-iota 6437  df-fun 6483  df-fv 6489  df-topgen 17344

This theorem is referenced by:  eltg4i  22873  eltg3i  22874  bastg  22879  tgss  22881  eltop  22887  tgqtop  23625  isfne4  36373