MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eltg Structured version   Visualization version   GIF version

Theorem eltg 22784
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 22782 . . 3 (𝐡 ∈ 𝑉 β†’ (topGenβ€˜π΅) = {π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯)})
21eleq2d 2811 . 2 (𝐡 ∈ 𝑉 β†’ (𝐴 ∈ (topGenβ€˜π΅) ↔ 𝐴 ∈ {π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯)}))
3 elex 3485 . . . 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 5310 . . . . . 6 (𝐡 ∈ 𝑉 β†’ (𝐡 ∩ 𝒫 𝐴) ∈ V)
65uniexd 7726 . . . . 5 (𝐡 ∈ 𝑉 β†’ βˆͺ (𝐡 ∩ 𝒫 𝐴) ∈ V)
7 ssexg 5314 . . . . 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 4609 . . . . . . 7 (π‘₯ = 𝐴 β†’ 𝒫 π‘₯ = 𝒫 𝐴)
1211ineq2d 4205 . . . . . 6 (π‘₯ = 𝐴 β†’ (𝐡 ∩ 𝒫 π‘₯) = (𝐡 ∩ 𝒫 𝐴))
1312unieqd 4913 . . . . 5 (π‘₯ = 𝐴 β†’ βˆͺ (𝐡 ∩ 𝒫 π‘₯) = βˆͺ (𝐡 ∩ 𝒫 𝐴))
1410, 13sseq12d 4008 . . . 4 (π‘₯ = 𝐴 β†’ (π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯) ↔ 𝐴 βŠ† βˆͺ (𝐡 ∩ 𝒫 𝐴)))
1514elabg 3659 . . 3 (𝐴 ∈ V β†’ (𝐴 ∈ {π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯)} ↔ 𝐴 βŠ† βˆͺ (𝐡 ∩ 𝒫 𝐴)))
164, 9, 15pm5.21nd 799 . 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 1533   ∈ wcel 2098  {cab 2701  Vcvv 3466   ∩ cin 3940   βŠ† wss 3941  π’« cpw 4595  βˆͺ cuni 4900  β€˜cfv 6534  topGenctg 17384
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fv 6542  df-topgen 17390
This theorem is referenced by:  eltg4i  22787  eltg3i  22788  bastg  22793  tgss  22795  eltop  22801  tgqtop  23540  isfne4  35716
  Copyright terms: Public domain W3C validator