Theorem tgval2 20760

Description: Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 20773) that (topGen‘𝐵) is indeed a topology (on ∪ 𝐵, see unitg 20771). See also tgval 20759 and tgval3 20767. (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 20759	. 2 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})
2		inss1 3833	. . . . . . . . 9 ⊢ (𝐵 ∩ 𝒫 𝑥) ⊆ 𝐵
3	2	unissi 4461	. . . . . . . 8 ⊢ ∪ (𝐵 ∩ 𝒫 𝑥) ⊆ ∪ 𝐵
4	3	sseli 3599	. . . . . . 7 ⊢ (𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥) → 𝑦 ∈ ∪ 𝐵)
5	4	pm4.71ri 665	. . . . . 6 ⊢ (𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ (𝑦 ∈ ∪ 𝐵 ∧ 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥)))
6	5	ralbii 2980	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ 𝐵 ∧ 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥)))
7		r19.26 3064	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ 𝐵 ∧ 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥)) ↔ (∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥)))
8	6, 7	bitri 264	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ (∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥)))
9		dfss3 3592	. . . 4 ⊢ (𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥))
10		dfss3 3592	. . . . 5 ⊢ (𝑥 ⊆ ∪ 𝐵 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ 𝐵)
11		elin 3796	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝐵 ∩ 𝒫 𝑥) ↔ (𝑧 ∈ 𝐵 ∧ 𝑧 ∈ 𝒫 𝑥))
12	11	anbi2i 730	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)) ↔ (𝑦 ∈ 𝑧 ∧ (𝑧 ∈ 𝐵 ∧ 𝑧 ∈ 𝒫 𝑥)))
13		an12 838	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑧 ∧ (𝑧 ∈ 𝐵 ∧ 𝑧 ∈ 𝒫 𝑥)) ↔ (𝑧 ∈ 𝐵 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥)))
14	12, 13	bitri 264	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)) ↔ (𝑧 ∈ 𝐵 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥)))
15	14	exbii 1774	. . . . . . . 8 ⊢ (∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)) ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥)))
16		eluni 4439	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (𝐵 ∩ 𝒫 𝑥)))
17		df-rex 2918	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥) ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥)))
18	15, 16, 17	3bitr4i 292	. . . . . . 7 ⊢ (𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥))
19		selpw 4165	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝒫 𝑥 ↔ 𝑧 ⊆ 𝑥)
20	19	anbi2i 730	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥) ↔ (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
21	20	rexbii 3041	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝒫 𝑥) ↔ ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
22	18, 21	bitr2i 265	. . . . . 6 ⊢ (∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥) ↔ 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥))
23	22	ralbii 2980	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥) ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ ∪ (𝐵 ∩ 𝒫 𝑥))
24	10, 23	anbi12i 733	. . . 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 292	. . 3 ⊢ (𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥) ↔ (𝑥 ⊆ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)))
26	25	abbii 2739	. 2 ⊢ {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} = {𝑥 ∣ (𝑥 ⊆ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))}
27	1, 26	syl6eq 2672	1 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ (𝑥 ⊆ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))})

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608  ∀wral 2912  ∃wrex 2913   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 4158  ∪ cuni 4436  ‘cfv 5888  topGenctg 16098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-topgen 16104

This theorem is referenced by:  eltg2  20762