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Theorem eltg4i 22907
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 6933 . . . 4 (𝐴 ∈ (topGen‘𝐵) → 𝐵 ∈ dom topGen)
2 eltg 22904 . . . 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 266 . 2 (𝐴 ∈ (topGen‘𝐵) → 𝐴 (𝐵 ∩ 𝒫 𝐴))
5 inss2 4228 . . . . 5 (𝐵 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴
65unissi 4918 . . . 4 (𝐵 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴
7 unipw 5452 . . . 4 𝒫 𝐴 = 𝐴
86, 7sseqtri 4013 . . 3 (𝐵 ∩ 𝒫 𝐴) ⊆ 𝐴
98a1i 11 . 2 (𝐴 ∈ (topGen‘𝐵) → (𝐵 ∩ 𝒫 𝐴) ⊆ 𝐴)
104, 9eqssd 3994 1 (𝐴 ∈ (topGen‘𝐵) → 𝐴 = (𝐵 ∩ 𝒫 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098  cin 3943  wss 3944  𝒫 cpw 4604   cuni 4909  dom cdm 5678  cfv 6549  topGenctg 17422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557  df-topgen 17428
This theorem is referenced by:  eltg3  22909  tgdom  22925  tgidm  22927  ontgval  36046
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