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Theorem eltg4i 22326
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 ∈ (topGenβ€˜π΅) β†’ 𝐴 = βˆͺ (𝐡 ∩ 𝒫 𝐴))

Proof of Theorem eltg4i
StepHypRef Expression
1 elfvdm 6880 . . . 4 (𝐴 ∈ (topGenβ€˜π΅) β†’ 𝐡 ∈ dom topGen)
2 eltg 22323 . . . 4 (𝐡 ∈ dom topGen β†’ (𝐴 ∈ (topGenβ€˜π΅) ↔ 𝐴 βŠ† βˆͺ (𝐡 ∩ 𝒫 𝐴)))
31, 2syl 17 . . 3 (𝐴 ∈ (topGenβ€˜π΅) β†’ (𝐴 ∈ (topGenβ€˜π΅) ↔ 𝐴 βŠ† βˆͺ (𝐡 ∩ 𝒫 𝐴)))
43ibi 267 . 2 (𝐴 ∈ (topGenβ€˜π΅) β†’ 𝐴 βŠ† βˆͺ (𝐡 ∩ 𝒫 𝐴))
5 inss2 4190 . . . . 5 (𝐡 ∩ 𝒫 𝐴) βŠ† 𝒫 𝐴
65unissi 4875 . . . 4 βˆͺ (𝐡 ∩ 𝒫 𝐴) βŠ† βˆͺ 𝒫 𝐴
7 unipw 5408 . . . 4 βˆͺ 𝒫 𝐴 = 𝐴
86, 7sseqtri 3981 . . 3 βˆͺ (𝐡 ∩ 𝒫 𝐴) βŠ† 𝐴
98a1i 11 . 2 (𝐴 ∈ (topGenβ€˜π΅) β†’ βˆͺ (𝐡 ∩ 𝒫 𝐴) βŠ† 𝐴)
104, 9eqssd 3962 1 (𝐴 ∈ (topGenβ€˜π΅) β†’ 𝐴 = βˆͺ (𝐡 ∩ 𝒫 𝐴))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1542   ∈ wcel 2107   ∩ cin 3910   βŠ† wss 3911  π’« cpw 4561  βˆͺ cuni 4866  dom cdm 5634  β€˜cfv 6497  topGenctg 17324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-topgen 17330
This theorem is referenced by:  eltg3  22328  tgdom  22344  tgidm  22346  ontgval  34949
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