Theorem eltg 13555

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 12711	. . 3 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})
2	1	eleq2d 2247	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 ∈ {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)}))
3		elex 2749	. . . 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 4140	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∩ 𝒫 𝐴) ∈ V)
6		uniexg 4440	. . . . . 6 ⊢ ((𝐵 ∩ 𝒫 𝐴) ∈ V → ∪ (𝐵 ∩ 𝒫 𝐴) ∈ V)
7	5, 6	syl 14	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → ∪ (𝐵 ∩ 𝒫 𝐴) ∈ V)
8		ssexg 4143	. . . . 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 3579	. . . . . . 7 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
13	12	ineq2d 3337	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝐴))
14	13	unieqd 3821	. . . . 5 ⊢ (𝑥 = 𝐴 → ∪ (𝐵 ∩ 𝒫 𝑥) = ∪ (𝐵 ∩ 𝒫 𝐴))
15	11, 14	sseq12d 3187	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴)))
16	15	elabg 2884	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} ↔ 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴)))
17	4, 10, 16	pm5.21nd 916	. 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 1353   ∈ wcel 2148  {cab 2163  Vcvv 2738   ∩ cin 3129   ⊆ wss 3130  𝒫 cpw 3576  ∪ cuni 3810  ‘cfv 5217  topGenctg 12703

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-iota 5179  df-fun 5219  df-fv 5225  df-topgen 12709

This theorem is referenced by:  eltg4i  13558  eltg3i  13559  bastg  13564  tgss  13566  eltop  13572