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Theorem eltg4i 20575
Description: An open set in a topology generated by a basis is the union of all basic open sets contained in it. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
eltg4i (𝐴 ∈ (topGen‘𝐵) → 𝐴 = (𝐵 ∩ 𝒫 𝐴))

Proof of Theorem eltg4i
StepHypRef Expression
1 elfvdm 6130 . . . 4 (𝐴 ∈ (topGen‘𝐵) → 𝐵 ∈ dom topGen)
2 eltg 20572 . . . 4 (𝐵 ∈ dom topGen → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 (𝐵 ∩ 𝒫 𝐴)))
31, 2syl 17 . . 3 (𝐴 ∈ (topGen‘𝐵) → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 (𝐵 ∩ 𝒫 𝐴)))
43ibi 255 . 2 (𝐴 ∈ (topGen‘𝐵) → 𝐴 (𝐵 ∩ 𝒫 𝐴))
5 inss2 3796 . . . . 5 (𝐵 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴
65unissi 4397 . . . 4 (𝐵 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴
7 unipw 4845 . . . 4 𝒫 𝐴 = 𝐴
86, 7sseqtri 3600 . . 3 (𝐵 ∩ 𝒫 𝐴) ⊆ 𝐴
98a1i 11 . 2 (𝐴 ∈ (topGen‘𝐵) → (𝐵 ∩ 𝒫 𝐴) ⊆ 𝐴)
104, 9eqssd 3585 1 (𝐴 ∈ (topGen‘𝐵) → 𝐴 = (𝐵 ∩ 𝒫 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195   = wceq 1475  wcel 1977  cin 3539  wss 3540  𝒫 cpw 4108   cuni 4372  dom cdm 5038  cfv 5804  topGenctg 15921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-topgen 15927
This theorem is referenced by:  eltg3  20577  tgdom  20593  tgidm  20595  ontgval  31600
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