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Theorem eltg 21657
 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 (𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 (𝐵 ∩ 𝒫 𝐴)))

Proof of Theorem eltg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 tgval 21655 . . 3 (𝐵𝑉 → (topGen‘𝐵) = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})
21eleq2d 2837 . 2 (𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 ∈ {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)}))
3 elex 3428 . . . 4 (𝐴 ∈ {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} → 𝐴 ∈ V)
43adantl 485 . . 3 ((𝐵𝑉𝐴 ∈ {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)}) → 𝐴 ∈ V)
5 inex1g 5189 . . . . . 6 (𝐵𝑉 → (𝐵 ∩ 𝒫 𝐴) ∈ V)
65uniexd 7466 . . . . 5 (𝐵𝑉 (𝐵 ∩ 𝒫 𝐴) ∈ V)
7 ssexg 5193 . . . . 5 ((𝐴 (𝐵 ∩ 𝒫 𝐴) ∧ (𝐵 ∩ 𝒫 𝐴) ∈ V) → 𝐴 ∈ V)
86, 7sylan2 595 . . . 4 ((𝐴 (𝐵 ∩ 𝒫 𝐴) ∧ 𝐵𝑉) → 𝐴 ∈ V)
98ancoms 462 . . 3 ((𝐵𝑉𝐴 (𝐵 ∩ 𝒫 𝐴)) → 𝐴 ∈ V)
10 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
11 pweq 4510 . . . . . . 7 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
1211ineq2d 4117 . . . . . 6 (𝑥 = 𝐴 → (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝐴))
1312unieqd 4812 . . . . 5 (𝑥 = 𝐴 (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝐴))
1410, 13sseq12d 3925 . . . 4 (𝑥 = 𝐴 → (𝑥 (𝐵 ∩ 𝒫 𝑥) ↔ 𝐴 (𝐵 ∩ 𝒫 𝐴)))
1514elabg 3587 . . 3 (𝐴 ∈ V → (𝐴 ∈ {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ↔ 𝐴 (𝐵 ∩ 𝒫 𝐴)))
164, 9, 15pm5.21nd 801 . 2 (𝐵𝑉 → (𝐴 ∈ {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ↔ 𝐴 (𝐵 ∩ 𝒫 𝐴)))
172, 16bitrd 282 1 (𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 (𝐵 ∩ 𝒫 𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  {cab 2735  Vcvv 3409   ∩ cin 3857   ⊆ wss 3858  𝒫 cpw 4494  ∪ cuni 4798  ‘cfv 6335  topGenctg 16769 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-iota 6294  df-fun 6337  df-fv 6343  df-topgen 16775 This theorem is referenced by:  eltg4i  21660  eltg3i  21661  bastg  21666  tgss  21668  eltop  21674  tgqtop  22412  isfne4  34078
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