Theorem eltg2 22214

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 22212	. . 3 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑧 ∣ (𝑧 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧))})
2	1	eleq2d 2823	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 ∈ {𝑧 ∣ (𝑧 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧))}))
3		elex 3460	. . . 4 ⊢ (𝐴 ∈ {𝑧 ∣ (𝑧 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧))} → 𝐴 ∈ V)
4	3	adantl 483	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ {𝑧 ∣ (𝑧 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧))}) → 𝐴 ∈ V)
5		uniexg 7660	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ∈ V)
6		ssexg 5272	. . . . . 6 ⊢ ((𝐴 ⊆ ∪ 𝐵 ∧ ∪ 𝐵 ∈ V) → 𝐴 ∈ V)
7	5, 6	sylan2 594	. . . . 5 ⊢ ((𝐴 ⊆ ∪ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V)
8	7	ancoms 460	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ ∪ 𝐵) → 𝐴 ∈ V)
9	8	adantrr 715	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐴 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))) → 𝐴 ∈ V)
10		sseq1 3961	. . . . 5 ⊢ (𝑧 = 𝐴 → (𝑧 ⊆ ∪ 𝐵 ↔ 𝐴 ⊆ ∪ 𝐵))
11		sseq2 3962	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ 𝐴))
12	11	anbi2d 630	. . . . . . 7 ⊢ (𝑧 = 𝐴 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴)))
13	12	rexbidv 3172	. . . . . 6 ⊢ (𝑧 = 𝐴 → (∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧) ↔ ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴)))
14	13	raleqbi1dv 3304	. . . . 5 ⊢ (𝑧 = 𝐴 → (∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴)))
15	10, 14	anbi12d 632	. . . 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 3621	. . 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 800	. 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 279	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ (𝐴 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1541   ∈ wcel 2106  {cab 2714  ∀wral 3062  ∃wrex 3071  Vcvv 3442   ⊆ wss 3902  ∪ cuni 4857  ‘cfv 6484  topGenctg 17246

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 2708  ax-sep 5248  ax-nul 5255  ax-pow 5313  ax-pr 5377  ax-un 7655

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3444  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4275  df-if 4479  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4858  df-br 5098  df-opab 5160  df-mpt 5181  df-id 5523  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6436  df-fun 6486  df-fv 6492  df-topgen 17252

This theorem is referenced by:  eltg2b  22215  tg1  22220  tgcl  22225  elmopn  23701  psmetutop  23829