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Theorem eltg 20955
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 20953 . . 3 (𝐵𝑉 → (topGen‘𝐵) = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})
21eleq2d 2817 . 2 (𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 ∈ {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)}))
3 elex 3344 . . . 4 (𝐴 ∈ {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} → 𝐴 ∈ V)
43adantl 473 . . 3 ((𝐵𝑉𝐴 ∈ {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)}) → 𝐴 ∈ V)
5 inex1g 4945 . . . . . 6 (𝐵𝑉 → (𝐵 ∩ 𝒫 𝐴) ∈ V)
6 uniexg 7112 . . . . . 6 ((𝐵 ∩ 𝒫 𝐴) ∈ V → (𝐵 ∩ 𝒫 𝐴) ∈ V)
75, 6syl 17 . . . . 5 (𝐵𝑉 (𝐵 ∩ 𝒫 𝐴) ∈ V)
8 ssexg 4948 . . . . 5 ((𝐴 (𝐵 ∩ 𝒫 𝐴) ∧ (𝐵 ∩ 𝒫 𝐴) ∈ V) → 𝐴 ∈ V)
97, 8sylan2 492 . . . 4 ((𝐴 (𝐵 ∩ 𝒫 𝐴) ∧ 𝐵𝑉) → 𝐴 ∈ V)
109ancoms 468 . . 3 ((𝐵𝑉𝐴 (𝐵 ∩ 𝒫 𝐴)) → 𝐴 ∈ V)
11 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
12 pweq 4297 . . . . . . 7 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
1312ineq2d 3949 . . . . . 6 (𝑥 = 𝐴 → (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝐴))
1413unieqd 4590 . . . . 5 (𝑥 = 𝐴 (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝐴))
1511, 14sseq12d 3767 . . . 4 (𝑥 = 𝐴 → (𝑥 (𝐵 ∩ 𝒫 𝑥) ↔ 𝐴 (𝐵 ∩ 𝒫 𝐴)))
1615elabg 3483 . . 3 (𝐴 ∈ V → (𝐴 ∈ {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ↔ 𝐴 (𝐵 ∩ 𝒫 𝐴)))
174, 10, 16pm5.21nd 979 . 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 1624  wcel 2131  {cab 2738  Vcvv 3332  cin 3706  wss 3707  𝒫 cpw 4294   cuni 4580  cfv 6041  topGenctg 16292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-iota 6004  df-fun 6043  df-fv 6049  df-topgen 16298
This theorem is referenced by:  eltg4i  20958  eltg3i  20959  bastg  20964  tgss  20966  eltop  20972  tgqtop  21709  isfne4  32633
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