Theorem eltg 22860

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 22858	. . 3 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})
2	1	eleq2d 2814	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 ∈ {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)}))
3		elex 3459	. . . 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 5261	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∩ 𝒫 𝐴) ∈ V)
6	5	uniexd 7682	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → ∪ (𝐵 ∩ 𝒫 𝐴) ∈ V)
7		ssexg 5265	. . . . 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 4567	. . . . . . 7 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
12	11	ineq2d 4173	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝐴))
13	12	unieqd 4874	. . . . 5 ⊢ (𝑥 = 𝐴 → ∪ (𝐵 ∩ 𝒫 𝑥) = ∪ (𝐵 ∩ 𝒫 𝐴))
14	10, 13	sseq12d 3971	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴)))
15	14	elabg 3634	. . 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 1540   ∈ wcel 2109  {cab 2707  Vcvv 3438   ∩ cin 3904   ⊆ wss 3905  𝒫 cpw 4553  ∪ cuni 4861  ‘cfv 6486  topGenctg 17359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-iota 6442  df-fun 6488  df-fv 6494  df-topgen 17365

This theorem is referenced by:  eltg4i  22863  eltg3i  22864  bastg  22869  tgss  22871  eltop  22877  tgqtop  23615  isfne4  36313