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Theorem eltg2 20573
 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 (𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ (𝐴 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝑦𝐴))))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝑉,𝑦

Proof of Theorem eltg2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 tgval2 20571 . . 3 (𝐵𝑉 → (topGen‘𝐵) = {𝑧 ∣ (𝑧 𝐵 ∧ ∀𝑥𝑧𝑦𝐵 (𝑥𝑦𝑦𝑧))})
21eleq2d 2673 . 2 (𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 ∈ {𝑧 ∣ (𝑧 𝐵 ∧ ∀𝑥𝑧𝑦𝐵 (𝑥𝑦𝑦𝑧))}))
3 elex 3185 . . . 4 (𝐴 ∈ {𝑧 ∣ (𝑧 𝐵 ∧ ∀𝑥𝑧𝑦𝐵 (𝑥𝑦𝑦𝑧))} → 𝐴 ∈ V)
43adantl 481 . . 3 ((𝐵𝑉𝐴 ∈ {𝑧 ∣ (𝑧 𝐵 ∧ ∀𝑥𝑧𝑦𝐵 (𝑥𝑦𝑦𝑧))}) → 𝐴 ∈ V)
5 uniexg 6853 . . . . . 6 (𝐵𝑉 𝐵 ∈ V)
6 ssexg 4732 . . . . . 6 ((𝐴 𝐵 𝐵 ∈ V) → 𝐴 ∈ V)
75, 6sylan2 490 . . . . 5 ((𝐴 𝐵𝐵𝑉) → 𝐴 ∈ V)
87ancoms 468 . . . 4 ((𝐵𝑉𝐴 𝐵) → 𝐴 ∈ V)
98adantrr 749 . . 3 ((𝐵𝑉 ∧ (𝐴 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝑦𝐴))) → 𝐴 ∈ V)
10 sseq1 3589 . . . . 5 (𝑧 = 𝐴 → (𝑧 𝐵𝐴 𝐵))
11 sseq2 3590 . . . . . . . 8 (𝑧 = 𝐴 → (𝑦𝑧𝑦𝐴))
1211anbi2d 736 . . . . . . 7 (𝑧 = 𝐴 → ((𝑥𝑦𝑦𝑧) ↔ (𝑥𝑦𝑦𝐴)))
1312rexbidv 3034 . . . . . 6 (𝑧 = 𝐴 → (∃𝑦𝐵 (𝑥𝑦𝑦𝑧) ↔ ∃𝑦𝐵 (𝑥𝑦𝑦𝐴)))
1413raleqbi1dv 3123 . . . . 5 (𝑧 = 𝐴 → (∀𝑥𝑧𝑦𝐵 (𝑥𝑦𝑦𝑧) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝑦𝐴)))
1510, 14anbi12d 743 . . . 4 (𝑧 = 𝐴 → ((𝑧 𝐵 ∧ ∀𝑥𝑧𝑦𝐵 (𝑥𝑦𝑦𝑧)) ↔ (𝐴 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝑦𝐴))))
1615elabg 3320 . . 3 (𝐴 ∈ V → (𝐴 ∈ {𝑧 ∣ (𝑧 𝐵 ∧ ∀𝑥𝑧𝑦𝐵 (𝑥𝑦𝑦𝑧))} ↔ (𝐴 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝑦𝐴))))
174, 9, 16pm5.21nd 939 . 2 (𝐵𝑉 → (𝐴 ∈ {𝑧 ∣ (𝑧 𝐵 ∧ ∀𝑥𝑧𝑦𝐵 (𝑥𝑦𝑦𝑧))} ↔ (𝐴 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝑦𝐴))))
182, 17bitrd 267 1 (𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ (𝐴 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝑦𝐴))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∪ cuni 4372  ‘cfv 5804  topGenctg 15921 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-topgen 15927 This theorem is referenced by:  eltg2b  20574  tg1  20579  tgcl  20584  elmopn  22057  psmetutop  22182
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