Theorem eltg 14568

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 13138	. . 3 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})
2	1	eleq2d 2276	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 ∈ {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)}))
3		elex 2784	. . . 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 4184	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∩ 𝒫 𝐴) ∈ V)
6		uniexg 4490	. . . . . 6 ⊢ ((𝐵 ∩ 𝒫 𝐴) ∈ V → ∪ (𝐵 ∩ 𝒫 𝐴) ∈ V)
7	5, 6	syl 14	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → ∪ (𝐵 ∩ 𝒫 𝐴) ∈ V)
8		ssexg 4187	. . . . 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 3620	. . . . . . 7 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
13	12	ineq2d 3375	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝐴))
14	13	unieqd 3863	. . . . 5 ⊢ (𝑥 = 𝐴 → ∪ (𝐵 ∩ 𝒫 𝑥) = ∪ (𝐵 ∩ 𝒫 𝐴))
15	11, 14	sseq12d 3225	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴)))
16	15	elabg 2920	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} ↔ 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴)))
17	4, 10, 16	pm5.21nd 918	. 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 1373   ∈ wcel 2177  {cab 2192  Vcvv 2773   ∩ cin 3166   ⊆ wss 3167  𝒫 cpw 3617  ∪ cuni 3852  ‘cfv 5276  topGenctg 13130

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3000  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-iota 5237  df-fun 5278  df-fv 5284  df-topgen 13136

This theorem is referenced by:  eltg4i  14571  eltg3i  14572  bastg  14577  tgss  14579  eltop  14585