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Theorem eltg 22442
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 22440 . . 3 (𝐵𝑉 → (topGen‘𝐵) = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})
21eleq2d 2820 . 2 (𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 ∈ {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)}))
3 elex 3493 . . . 4 (𝐴 ∈ {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} → 𝐴 ∈ V)
43adantl 483 . . 3 ((𝐵𝑉𝐴 ∈ {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)}) → 𝐴 ∈ V)
5 inex1g 5318 . . . . . 6 (𝐵𝑉 → (𝐵 ∩ 𝒫 𝐴) ∈ V)
65uniexd 7727 . . . . 5 (𝐵𝑉 (𝐵 ∩ 𝒫 𝐴) ∈ V)
7 ssexg 5322 . . . . 5 ((𝐴 (𝐵 ∩ 𝒫 𝐴) ∧ (𝐵 ∩ 𝒫 𝐴) ∈ V) → 𝐴 ∈ V)
86, 7sylan2 594 . . . 4 ((𝐴 (𝐵 ∩ 𝒫 𝐴) ∧ 𝐵𝑉) → 𝐴 ∈ V)
98ancoms 460 . . 3 ((𝐵𝑉𝐴 (𝐵 ∩ 𝒫 𝐴)) → 𝐴 ∈ V)
10 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
11 pweq 4615 . . . . . . 7 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
1211ineq2d 4211 . . . . . 6 (𝑥 = 𝐴 → (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝐴))
1312unieqd 4921 . . . . 5 (𝑥 = 𝐴 (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝐴))
1410, 13sseq12d 4014 . . . 4 (𝑥 = 𝐴 → (𝑥 (𝐵 ∩ 𝒫 𝑥) ↔ 𝐴 (𝐵 ∩ 𝒫 𝐴)))
1514elabg 3665 . . 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 279 1 (𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 (𝐵 ∩ 𝒫 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  {cab 2710  Vcvv 3475  cin 3946  wss 3947  𝒫 cpw 4601   cuni 4907  cfv 6540  topGenctg 17379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-topgen 17385
This theorem is referenced by:  eltg4i  22445  eltg3i  22446  bastg  22451  tgss  22453  eltop  22459  tgqtop  23198  isfne4  35163
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