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Theorem eltg 20667
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 20665 . . 3 (𝐵𝑉 → (topGen‘𝐵) = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})
21eleq2d 2689 . 2 (𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 ∈ {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)}))
3 elex 3203 . . . 4 (𝐴 ∈ {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} → 𝐴 ∈ V)
43adantl 482 . . 3 ((𝐵𝑉𝐴 ∈ {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)}) → 𝐴 ∈ V)
5 inex1g 4766 . . . . . 6 (𝐵𝑉 → (𝐵 ∩ 𝒫 𝐴) ∈ V)
6 uniexg 6909 . . . . . 6 ((𝐵 ∩ 𝒫 𝐴) ∈ V → (𝐵 ∩ 𝒫 𝐴) ∈ V)
75, 6syl 17 . . . . 5 (𝐵𝑉 (𝐵 ∩ 𝒫 𝐴) ∈ V)
8 ssexg 4769 . . . . 5 ((𝐴 (𝐵 ∩ 𝒫 𝐴) ∧ (𝐵 ∩ 𝒫 𝐴) ∈ V) → 𝐴 ∈ V)
97, 8sylan2 491 . . . 4 ((𝐴 (𝐵 ∩ 𝒫 𝐴) ∧ 𝐵𝑉) → 𝐴 ∈ V)
109ancoms 469 . . 3 ((𝐵𝑉𝐴 (𝐵 ∩ 𝒫 𝐴)) → 𝐴 ∈ V)
11 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
12 pweq 4138 . . . . . . 7 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
1312ineq2d 3797 . . . . . 6 (𝑥 = 𝐴 → (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝐴))
1413unieqd 4417 . . . . 5 (𝑥 = 𝐴 (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝐴))
1511, 14sseq12d 3618 . . . 4 (𝑥 = 𝐴 → (𝑥 (𝐵 ∩ 𝒫 𝑥) ↔ 𝐴 (𝐵 ∩ 𝒫 𝐴)))
1615elabg 3339 . . 3 (𝐴 ∈ V → (𝐴 ∈ {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ↔ 𝐴 (𝐵 ∩ 𝒫 𝐴)))
174, 10, 16pm5.21nd 940 . 2 (𝐵𝑉 → (𝐴 ∈ {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ↔ 𝐴 (𝐵 ∩ 𝒫 𝐴)))
182, 17bitrd 268 1 (𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 (𝐵 ∩ 𝒫 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wcel 1992  {cab 2612  Vcvv 3191  cin 3559  wss 3560  𝒫 cpw 4135   cuni 4407  cfv 5850  topGenctg 16014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fun 5852  df-fv 5858  df-topgen 16020
This theorem is referenced by:  eltg4i  20670  eltg3i  20671  bastg  20676  tgss  20678  eltop  20684  tgqtop  21420  isfne4  31950
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