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Theorem eltg 22853
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 22851 . . 3 (𝐡 ∈ 𝑉 β†’ (topGenβ€˜π΅) = {π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯)})
21eleq2d 2815 . 2 (𝐡 ∈ 𝑉 β†’ (𝐴 ∈ (topGenβ€˜π΅) ↔ 𝐴 ∈ {π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯)}))
3 elex 3489 . . . 4 (𝐴 ∈ {π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯)} β†’ 𝐴 ∈ V)
43adantl 481 . . 3 ((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ {π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯)}) β†’ 𝐴 ∈ V)
5 inex1g 5313 . . . . . 6 (𝐡 ∈ 𝑉 β†’ (𝐡 ∩ 𝒫 𝐴) ∈ V)
65uniexd 7741 . . . . 5 (𝐡 ∈ 𝑉 β†’ βˆͺ (𝐡 ∩ 𝒫 𝐴) ∈ V)
7 ssexg 5317 . . . . 5 ((𝐴 βŠ† βˆͺ (𝐡 ∩ 𝒫 𝐴) ∧ βˆͺ (𝐡 ∩ 𝒫 𝐴) ∈ V) β†’ 𝐴 ∈ V)
86, 7sylan2 592 . . . 4 ((𝐴 βŠ† βˆͺ (𝐡 ∩ 𝒫 𝐴) ∧ 𝐡 ∈ 𝑉) β†’ 𝐴 ∈ V)
98ancoms 458 . . 3 ((𝐡 ∈ 𝑉 ∧ 𝐴 βŠ† βˆͺ (𝐡 ∩ 𝒫 𝐴)) β†’ 𝐴 ∈ V)
10 id 22 . . . . 5 (π‘₯ = 𝐴 β†’ π‘₯ = 𝐴)
11 pweq 4612 . . . . . . 7 (π‘₯ = 𝐴 β†’ 𝒫 π‘₯ = 𝒫 𝐴)
1211ineq2d 4208 . . . . . 6 (π‘₯ = 𝐴 β†’ (𝐡 ∩ 𝒫 π‘₯) = (𝐡 ∩ 𝒫 𝐴))
1312unieqd 4916 . . . . 5 (π‘₯ = 𝐴 β†’ βˆͺ (𝐡 ∩ 𝒫 π‘₯) = βˆͺ (𝐡 ∩ 𝒫 𝐴))
1410, 13sseq12d 4011 . . . 4 (π‘₯ = 𝐴 β†’ (π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯) ↔ 𝐴 βŠ† βˆͺ (𝐡 ∩ 𝒫 𝐴)))
1514elabg 3664 . . 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 1534   ∈ wcel 2099  {cab 2705  Vcvv 3470   ∩ cin 3944   βŠ† wss 3945  π’« cpw 4598  βˆͺ cuni 4903  β€˜cfv 6542  topGenctg 17412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-topgen 17418
This theorem is referenced by:  eltg4i  22856  eltg3i  22857  bastg  22862  tgss  22864  eltop  22870  tgqtop  23609  isfne4  35818
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