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Theorem eltg4i 21560
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 6695 . . . 4 (𝐴 ∈ (topGen‘𝐵) → 𝐵 ∈ dom topGen)
2 eltg 21557 . . . 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 269 . 2 (𝐴 ∈ (topGen‘𝐵) → 𝐴 (𝐵 ∩ 𝒫 𝐴))
5 inss2 4204 . . . . 5 (𝐵 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴
65unissi 4853 . . . 4 (𝐵 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴
7 unipw 5333 . . . 4 𝒫 𝐴 = 𝐴
86, 7sseqtri 4001 . . 3 (𝐵 ∩ 𝒫 𝐴) ⊆ 𝐴
98a1i 11 . 2 (𝐴 ∈ (topGen‘𝐵) → (𝐵 ∩ 𝒫 𝐴) ⊆ 𝐴)
104, 9eqssd 3982 1 (𝐴 ∈ (topGen‘𝐵) → 𝐴 = (𝐵 ∩ 𝒫 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208   = wceq 1530  wcel 2107  cin 3933  wss 3934  𝒫 cpw 4537   cuni 4830  dom cdm 5548  cfv 6348  topGenctg 16703
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-topgen 16709
This theorem is referenced by:  eltg3  21562  tgdom  21578  tgidm  21580  ontgval  33772
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