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Theorem eltg 22323
Description: Membership in a topology generated by a basis. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
Assertion
Ref Expression
eltg (𝐡 ∈ 𝑉 β†’ (𝐴 ∈ (topGenβ€˜π΅) ↔ 𝐴 βŠ† βˆͺ (𝐡 ∩ 𝒫 𝐴)))

Proof of Theorem eltg
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 tgval 22321 . . 3 (𝐡 ∈ 𝑉 β†’ (topGenβ€˜π΅) = {π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯)})
21eleq2d 2820 . 2 (𝐡 ∈ 𝑉 β†’ (𝐴 ∈ (topGenβ€˜π΅) ↔ 𝐴 ∈ {π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯)}))
3 elex 3462 . . . 4 (𝐴 ∈ {π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯)} β†’ 𝐴 ∈ V)
43adantl 483 . . 3 ((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ {π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯)}) β†’ 𝐴 ∈ V)
5 inex1g 5277 . . . . . 6 (𝐡 ∈ 𝑉 β†’ (𝐡 ∩ 𝒫 𝐴) ∈ V)
65uniexd 7680 . . . . 5 (𝐡 ∈ 𝑉 β†’ βˆͺ (𝐡 ∩ 𝒫 𝐴) ∈ V)
7 ssexg 5281 . . . . 5 ((𝐴 βŠ† βˆͺ (𝐡 ∩ 𝒫 𝐴) ∧ βˆͺ (𝐡 ∩ 𝒫 𝐴) ∈ V) β†’ 𝐴 ∈ V)
86, 7sylan2 594 . . . 4 ((𝐴 βŠ† βˆͺ (𝐡 ∩ 𝒫 𝐴) ∧ 𝐡 ∈ 𝑉) β†’ 𝐴 ∈ V)
98ancoms 460 . . 3 ((𝐡 ∈ 𝑉 ∧ 𝐴 βŠ† βˆͺ (𝐡 ∩ 𝒫 𝐴)) β†’ 𝐴 ∈ V)
10 id 22 . . . . 5 (π‘₯ = 𝐴 β†’ π‘₯ = 𝐴)
11 pweq 4575 . . . . . . 7 (π‘₯ = 𝐴 β†’ 𝒫 π‘₯ = 𝒫 𝐴)
1211ineq2d 4173 . . . . . 6 (π‘₯ = 𝐴 β†’ (𝐡 ∩ 𝒫 π‘₯) = (𝐡 ∩ 𝒫 𝐴))
1312unieqd 4880 . . . . 5 (π‘₯ = 𝐴 β†’ βˆͺ (𝐡 ∩ 𝒫 π‘₯) = βˆͺ (𝐡 ∩ 𝒫 𝐴))
1410, 13sseq12d 3978 . . . 4 (π‘₯ = 𝐴 β†’ (π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯) ↔ 𝐴 βŠ† βˆͺ (𝐡 ∩ 𝒫 𝐴)))
1514elabg 3629 . . 3 (𝐴 ∈ V β†’ (𝐴 ∈ {π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯)} ↔ 𝐴 βŠ† βˆͺ (𝐡 ∩ 𝒫 𝐴)))
164, 9, 15pm5.21nd 801 . 2 (𝐡 ∈ 𝑉 β†’ (𝐴 ∈ {π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯)} ↔ 𝐴 βŠ† βˆͺ (𝐡 ∩ 𝒫 𝐴)))
172, 16bitrd 279 1 (𝐡 ∈ 𝑉 β†’ (𝐴 ∈ (topGenβ€˜π΅) ↔ 𝐴 βŠ† βˆͺ (𝐡 ∩ 𝒫 𝐴)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1542   ∈ wcel 2107  {cab 2710  Vcvv 3444   ∩ cin 3910   βŠ† wss 3911  π’« cpw 4561  βˆͺ cuni 4866  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:  eltg4i  22326  eltg3i  22327  bastg  22332  tgss  22334  eltop  22340  tgqtop  23079  isfne4  34858
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