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Theorem eltg2 22860
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 22858 . . 3 (𝐡 ∈ 𝑉 β†’ (topGenβ€˜π΅) = {𝑧 ∣ (𝑧 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝑧 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝑧))})
21eleq2d 2815 . 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 3490 . . . 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 7745 . . . . . 6 (𝐡 ∈ 𝑉 β†’ βˆͺ 𝐡 ∈ V)
6 ssexg 5323 . . . . . 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 716 . . 3 ((𝐡 ∈ 𝑉 ∧ (𝐴 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴))) β†’ 𝐴 ∈ V)
10 sseq1 4005 . . . . 5 (𝑧 = 𝐴 β†’ (𝑧 βŠ† βˆͺ 𝐡 ↔ 𝐴 βŠ† βˆͺ 𝐡))
11 sseq2 4006 . . . . . . . 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 3175 . . . . . 6 (𝑧 = 𝐴 β†’ (βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝑧) ↔ βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴)))
1413raleqbi1dv 3330 . . . . 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 3665 . . 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 801 . 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 1534   ∈ wcel 2099  {cab 2705  βˆ€wral 3058  βˆƒwrex 3067  Vcvv 3471   βŠ† wss 3947  βˆͺ cuni 4908  β€˜cfv 6548  topGenctg 17418
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 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
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 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-topgen 17424
This theorem is referenced by:  eltg2b  22861  tg1  22866  tgcl  22871  elmopn  24347  psmetutop  24475
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