Theorem tgval2 14774

Description: Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 14787) that (topGen‘𝐵) is indeed a topology (on ∪ 𝐵, see unitg 14785). See also tgval 13344 and tgval3 14781. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)

Assertion

Ref	Expression
tgval2	⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ (𝑥 ⊆ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))})

Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥,𝑉,𝑦,𝑧

Proof of Theorem tgval2

Step	Hyp	Ref	Expression
1		tgval 13344	. 2 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})
2		inss1 3427	. . . . . . . . 9 ⊢ (𝐵 ∩ 𝒫 𝑥) ⊆ 𝐵
3	2	unissi 3916	. . . . . . . 8 ⊢ ∪ (𝐵 ∩ 𝒫 𝑥) ⊆ ∪ 𝐵
4	3	sseli 3223	. . . . . . 7 ⊢ (𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥) → 𝑦 ∈ ∪ 𝐵)
5	4	pm4.71ri 392	. . . . . 6 ⊢ (𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ (𝑦 ∈ ∪ 𝐵 ∧ 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥)))
6	5	ralbii 2538	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ 𝐵 ∧ 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥)))
7		r19.26 2659	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ 𝐵 ∧ 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥)) ↔ (∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥)))
8	6, 7	bitri 184	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ (∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥)))
9		dfss3 3216	. . . 4 ⊢ (𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥))
10		dfss3 3216	. . . . 5 ⊢ (𝑥 ⊆ ∪ 𝐵 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ 𝐵)
11		elin 3390	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝐵 ∩ 𝒫 𝑥) ↔ (𝑧 ∈ 𝐵 ∧ 𝑧 ∈ 𝒫 𝑥))
12	11	anbi2i 457	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)) ↔ (𝑦 ∈ 𝑧 ∧ (𝑧 ∈ 𝐵 ∧ 𝑧 ∈ 𝒫 𝑥)))
13		an12 563	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑧 ∧ (𝑧 ∈ 𝐵 ∧ 𝑧 ∈ 𝒫 𝑥)) ↔ (𝑧 ∈ 𝐵 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥)))
14	12, 13	bitri 184	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)) ↔ (𝑧 ∈ 𝐵 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥)))
15	14	exbii 1653	. . . . . . . 8 ⊢ (∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)) ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥)))
16		eluni 3896	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)))
17		df-rex 2516	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥) ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥)))
18	15, 16, 17	3bitr4i 212	. . . . . . 7 ⊢ (𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥))
19		velpw 3659	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝒫 𝑥 ↔ 𝑧 ⊆ 𝑥)
20	19	anbi2i 457	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥) ↔ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
21	20	rexbii 2539	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥) ↔ ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
22	18, 21	bitr2i 185	. . . . . 6 ⊢ (∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥) ↔ 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥))
23	22	ralbii 2538	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥) ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥))
24	10, 23	anbi12i 460	. . . 4 ⊢ ((𝑥 ⊆ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)) ↔ (∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥)))
25	8, 9, 24	3bitr4i 212	. . 3 ⊢ (𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ (𝑥 ⊆ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)))
26	25	abbii 2347	. 2 ⊢ {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} = {𝑥 ∣ (𝑥 ⊆ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))}
27	1, 26	eqtrdi 2280	1 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ (𝑥 ⊆ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397  ∃wex 1540   ∈ wcel 2202  {cab 2217  ∀wral 2510  ∃wrex 2511   ∩ cin 3199   ⊆ wss 3200  𝒫 cpw 3652  ∪ cuni 3893  ‘cfv 5326  topGenctg 13336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-topgen 13342

This theorem is referenced by:  eltg2  14776