Theorem eltg2 22089

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	⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ (𝐴 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))))

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.

Step	Hyp	Ref	Expression
1		tgval2 22087	. . 3 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑧 ∣ (𝑧 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧))})
2	1	eleq2d 2825	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 ∈ {𝑧 ∣ (𝑧 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧))}))
3		elex 3448	. . . 4 ⊢ (𝐴 ∈ {𝑧 ∣ (𝑧 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧))} → 𝐴 ∈ V)
4	3	adantl 481	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ {𝑧 ∣ (𝑧 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧))}) → 𝐴 ∈ V)
5		uniexg 7584	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ∈ V)
6		ssexg 5250	. . . . . 6 ⊢ ((𝐴 ⊆ ∪ 𝐵 ∧ ∪ 𝐵 ∈ V) → 𝐴 ∈ V)
7	5, 6	sylan2 592	. . . . 5 ⊢ ((𝐴 ⊆ ∪ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V)
8	7	ancoms 458	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ ∪ 𝐵) → 𝐴 ∈ V)
9	8	adantrr 713	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐴 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))) → 𝐴 ∈ V)
10		sseq1 3950	. . . . 5 ⊢ (𝑧 = 𝐴 → (𝑧 ⊆ ∪ 𝐵 ↔ 𝐴 ⊆ ∪ 𝐵))
11		sseq2 3951	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ 𝐴))
12	11	anbi2d 628	. . . . . . 7 ⊢ (𝑧 = 𝐴 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴)))
13	12	rexbidv 3227	. . . . . 6 ⊢ (𝑧 = 𝐴 → (∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧) ↔ ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴)))
14	13	raleqbi1dv 3338	. . . . 5 ⊢ (𝑧 = 𝐴 → (∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴)))
15	10, 14	anbi12d 630	. . . 4 ⊢ (𝑧 = 𝐴 → ((𝑧 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧)) ↔ (𝐴 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))))
16	15	elabg 3608	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝑧 ∣ (𝑧 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧))} ↔ (𝐴 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))))
17	4, 9, 16	pm5.21nd 798	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ {𝑧 ∣ (𝑧 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧))} ↔ (𝐴 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))))
18	2, 17	bitrd 278	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ (𝐴 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1541   ∈ wcel 2109  {cab 2716  ∀wral 3065  ∃wrex 3066  Vcvv 3430   ⊆ wss 3891  ∪ cuni 4844  ‘cfv 6430  topGenctg 17129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-iota 6388  df-fun 6432  df-fv 6438  df-topgen 17135

This theorem is referenced by:  eltg2b  22090  tg1  22095  tgcl  22100  elmopn  23576  psmetutop  23704