Theorem eltg 14691

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 13261	. . 3 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})
2	1	eleq2d 2279	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 ∈ {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)}))
3		elex 2791	. . . 4 ⊢ (𝐴 ∈ {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} → 𝐴 ∈ V)
4	3	adantl 277	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)}) → 𝐴 ∈ V)
5		inex1g 4199	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∩ 𝒫 𝐴) ∈ V)
6		uniexg 4507	. . . . . 6 ⊢ ((𝐵 ∩ 𝒫 𝐴) ∈ V → ∪ (𝐵 ∩ 𝒫 𝐴) ∈ V)
7	5, 6	syl 14	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → ∪ (𝐵 ∩ 𝒫 𝐴) ∈ V)
8		ssexg 4202	. . . . 5 ⊢ ((𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴) ∧ ∪ (𝐵 ∩ 𝒫 𝐴) ∈ V) → 𝐴 ∈ V)
9	7, 8	sylan2 286	. . . 4 ⊢ ((𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴) ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V)
10	9	ancoms 268	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴)) → 𝐴 ∈ V)
11		id 19	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
12		pweq 3632	. . . . . . 7 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
13	12	ineq2d 3385	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝐴))
14	13	unieqd 3878	. . . . 5 ⊢ (𝑥 = 𝐴 → ∪ (𝐵 ∩ 𝒫 𝑥) = ∪ (𝐵 ∩ 𝒫 𝐴))
15	11, 14	sseq12d 3235	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴)))
16	15	elabg 2929	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} ↔ 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴)))
17	4, 10, 16	pm5.21nd 920	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} ↔ 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴)))
18	2, 17	bitrd 188	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴)))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1375   ∈ wcel 2180  {cab 2195  Vcvv 2779   ∩ cin 3176   ⊆ wss 3177  𝒫 cpw 3629  ∪ cuni 3867  ‘cfv 5294  topGenctg 13253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-v 2781  df-sbc 3009  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-iota 5254  df-fun 5296  df-fv 5302  df-topgen 13259

This theorem is referenced by:  eltg4i  14694  eltg3i  14695  bastg  14700  tgss  14702  eltop  14708