Theorem eltg2 22902

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 22900	. . 3 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑧 ∣ (𝑧 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧))})
2	1	eleq2d 2822	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 ∈ {𝑧 ∣ (𝑧 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧))}))
3		elex 3461	. . . 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 7685	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ∈ V)
6		ssexg 5268	. . . . . 6 ⊢ ((𝐴 ⊆ ∪ 𝐵 ∧ ∪ 𝐵 ∈ V) → 𝐴 ∈ V)
7	5, 6	sylan2 593	. . . . 5 ⊢ ((𝐴 ⊆ ∪ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V)
8	7	ancoms 458	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ ∪ 𝐵) → 𝐴 ∈ V)
9	8	adantrr 717	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐴 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))) → 𝐴 ∈ V)
10		sseq1 3959	. . . . 5 ⊢ (𝑧 = 𝐴 → (𝑧 ⊆ ∪ 𝐵 ↔ 𝐴 ⊆ ∪ 𝐵))
11		sseq2 3960	. . . . . . . 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 3160	. . . . . 6 ⊢ (𝑧 = 𝐴 → (∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧) ↔ ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴)))
14	13	raleqbi1dv 3308	. . . . 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 3631	. . 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 801	. 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 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2714  ∀wral 3051  ∃wrex 3060  Vcvv 3440   ⊆ wss 3901  ∪ cuni 4863  ‘cfv 6492  topGenctg 17357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-topgen 17363

This theorem is referenced by:  eltg2b  22903  tg1  22908  tgcl  22913  elmopn  24386  psmetutop  24511