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Theorem eltg2 22460
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 22458 . . 3 (𝐡 ∈ 𝑉 β†’ (topGenβ€˜π΅) = {𝑧 ∣ (𝑧 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝑧 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝑧))})
21eleq2d 2819 . 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 3492 . . . 4 (𝐴 ∈ {𝑧 ∣ (𝑧 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝑧 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝑧))} β†’ 𝐴 ∈ V)
43adantl 482 . . 3 ((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ {𝑧 ∣ (𝑧 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝑧 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝑧))}) β†’ 𝐴 ∈ V)
5 uniexg 7729 . . . . . 6 (𝐡 ∈ 𝑉 β†’ βˆͺ 𝐡 ∈ V)
6 ssexg 5323 . . . . . 6 ((𝐴 βŠ† βˆͺ 𝐡 ∧ βˆͺ 𝐡 ∈ V) β†’ 𝐴 ∈ V)
75, 6sylan2 593 . . . . 5 ((𝐴 βŠ† βˆͺ 𝐡 ∧ 𝐡 ∈ 𝑉) β†’ 𝐴 ∈ V)
87ancoms 459 . . . 4 ((𝐡 ∈ 𝑉 ∧ 𝐴 βŠ† βˆͺ 𝐡) β†’ 𝐴 ∈ V)
98adantrr 715 . . 3 ((𝐡 ∈ 𝑉 ∧ (𝐴 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴))) β†’ 𝐴 ∈ V)
10 sseq1 4007 . . . . 5 (𝑧 = 𝐴 β†’ (𝑧 βŠ† βˆͺ 𝐡 ↔ 𝐴 βŠ† βˆͺ 𝐡))
11 sseq2 4008 . . . . . . . 8 (𝑧 = 𝐴 β†’ (𝑦 βŠ† 𝑧 ↔ 𝑦 βŠ† 𝐴))
1211anbi2d 629 . . . . . . 7 (𝑧 = 𝐴 β†’ ((π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝑧) ↔ (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴)))
1312rexbidv 3178 . . . . . 6 (𝑧 = 𝐴 β†’ (βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝑧) ↔ βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴)))
1413raleqbi1dv 3333 . . . . 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 631 . . . 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 3666 . . 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 800 . 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 278 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 396   = wceq 1541   ∈ wcel 2106  {cab 2709  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3474   βŠ† wss 3948  βˆͺ cuni 4908  β€˜cfv 6543  topGenctg 17382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-topgen 17388
This theorem is referenced by:  eltg2b  22461  tg1  22466  tgcl  22471  elmopn  23947  psmetutop  24075
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