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Theorem eltg2 22805
Description: Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
Assertion
Ref Expression
eltg2 (𝐡 ∈ 𝑉 β†’ (𝐴 ∈ (topGenβ€˜π΅) ↔ (𝐴 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴))))
Distinct variable groups:   π‘₯,𝑦,𝐴   π‘₯,𝐡,𝑦   π‘₯,𝑉,𝑦
Proof of Theorem eltg2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 tgval2 22803 . . 3 (𝐡 ∈ 𝑉 β†’ (topGenβ€˜π΅) = {𝑧 ∣ (𝑧 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝑧 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝑧))})
21eleq2d 2811 . 2 (𝐡 ∈ 𝑉 β†’ (𝐴 ∈ (topGenβ€˜π΅) ↔ 𝐴 ∈ {𝑧 ∣ (𝑧 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝑧 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝑧))}))
3 elex 3485 . . . 4 (𝐴 ∈ {𝑧 ∣ (𝑧 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝑧 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝑧))} β†’ 𝐴 ∈ V)
43adantl 481 . . 3 ((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ {𝑧 ∣ (𝑧 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝑧 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝑧))}) β†’ 𝐴 ∈ V)
5 uniexg 7724 . . . . . 6 (𝐡 ∈ 𝑉 β†’ βˆͺ 𝐡 ∈ V)
6 ssexg 5314 . . . . . 6 ((𝐴 βŠ† βˆͺ 𝐡 ∧ βˆͺ 𝐡 ∈ V) β†’ 𝐴 ∈ V)
75, 6sylan2 592 . . . . 5 ((𝐴 βŠ† βˆͺ 𝐡 ∧ 𝐡 ∈ 𝑉) β†’ 𝐴 ∈ V)
87ancoms 458 . . . 4 ((𝐡 ∈ 𝑉 ∧ 𝐴 βŠ† βˆͺ 𝐡) β†’ 𝐴 ∈ V)
98adantrr 714 . . 3 ((𝐡 ∈ 𝑉 ∧ (𝐴 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴))) β†’ 𝐴 ∈ V)
10 sseq1 4000 . . . . 5 (𝑧 = 𝐴 β†’ (𝑧 βŠ† βˆͺ 𝐡 ↔ 𝐴 βŠ† βˆͺ 𝐡))
11 sseq2 4001 . . . . . . . 8 (𝑧 = 𝐴 β†’ (𝑦 βŠ† 𝑧 ↔ 𝑦 βŠ† 𝐴))
1211anbi2d 628 . . . . . . 7 (𝑧 = 𝐴 β†’ ((π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝑧) ↔ (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴)))
1312rexbidv 3170 . . . . . 6 (𝑧 = 𝐴 β†’ (βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝑧) ↔ βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴)))
1413raleqbi1dv 3325 . . . . 5 (𝑧 = 𝐴 β†’ (βˆ€π‘₯ ∈ 𝑧 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝑧) ↔ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴)))
1510, 14anbi12d 630 . . . 4 (𝑧 = 𝐴 β†’ ((𝑧 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝑧 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝑧)) ↔ (𝐴 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴))))
1615elabg 3659 . . 3 (𝐴 ∈ V β†’ (𝐴 ∈ {𝑧 ∣ (𝑧 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝑧 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝑧))} ↔ (𝐴 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴))))
174, 9, 16pm5.21nd 799 . 2 (𝐡 ∈ 𝑉 β†’ (𝐴 ∈ {𝑧 ∣ (𝑧 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝑧 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝑧))} ↔ (𝐴 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴))))
182, 17bitrd 279 1 (𝐡 ∈ 𝑉 β†’ (𝐴 ∈ (topGenβ€˜π΅) ↔ (𝐴 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {cab 2701  βˆ€wral 3053  βˆƒwrex 3062  Vcvv 3466   βŠ† wss 3941  βˆͺ cuni 4900  β€˜cfv 6534  topGenctg 17388
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 17394
This theorem is referenced by:  eltg2b  22806  tg1  22811  tgcl  22816  elmopn  24292  psmetutop  24420
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