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Theorem eltg4i 20984
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 6361 . . . 4 (𝐴 ∈ (topGen‘𝐵) → 𝐵 ∈ dom topGen)
2 eltg 20981 . . . 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 256 . 2 (𝐴 ∈ (topGen‘𝐵) → 𝐴 (𝐵 ∩ 𝒫 𝐴))
5 inss2 3980 . . . . 5 (𝐵 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴
65unissi 4595 . . . 4 (𝐵 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴
7 unipw 5046 . . . 4 𝒫 𝐴 = 𝐴
86, 7sseqtri 3784 . . 3 (𝐵 ∩ 𝒫 𝐴) ⊆ 𝐴
98a1i 11 . 2 (𝐴 ∈ (topGen‘𝐵) → (𝐵 ∩ 𝒫 𝐴) ⊆ 𝐴)
104, 9eqssd 3767 1 (𝐴 ∈ (topGen‘𝐵) → 𝐴 = (𝐵 ∩ 𝒫 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1630  wcel 2144  cin 3720  wss 3721  𝒫 cpw 4295   cuni 4572  dom cdm 5249  cfv 6031  topGenctg 16305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-topgen 16311
This theorem is referenced by:  eltg3  20986  tgdom  21002  tgidm  21004  ontgval  32761
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