Theorem tgval 23180

Description: The topology generated by a basis. See also tgval2 23181 and tgval3 23188. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)

Assertion

Ref	Expression
tgval	⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})

Distinct variable groups:   𝑥,𝐵   𝑥,𝑉

Proof of Theorem tgval

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-topgen 17528	. 2 ⊢ topGen = (𝑦 ∈ V ↦ {𝑥 ∣ 𝑥 ⊆ ∪ (𝑦 ∩ 𝒫 𝑥)})
2		ineq1 4159	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑥))
3	2	unieqd 4880	. . . 4 ⊢ (𝑦 = 𝐵 → ∪ (𝑦 ∩ 𝒫 𝑥) = ∪ (𝐵 ∩ 𝒫 𝑥))
4	3	sseq2d 3963	. . 3 ⊢ (𝑦 = 𝐵 → (𝑥 ⊆ ∪ (𝑦 ∩ 𝒫 𝑥) ↔ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)))
5	4	abbidv 2826	. 2 ⊢ (𝑦 = 𝐵 → {𝑥 ∣ 𝑥 ⊆ ∪ (𝑦 ∩ 𝒫 𝑥)} = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})
6		elex 3471	. 2 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V)
7		uniexg 7742	. . 3 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ∈ V)
8		abssexg 5347	. . 3 ⊢ (∪ 𝐵 ∈ V → {𝑥 ∣ (𝑥 ⊆ ∪ 𝐵 ∧ 𝑥 ⊆ ∪ 𝒫 𝑥)} ∈ V)
9		uniin 4891	. . . . . . 7 ⊢ ∪ (𝐵 ∩ 𝒫 𝑥) ⊆ (∪ 𝐵 ∩ ∪ 𝒫 𝑥)
10		sstr 3939	. . . . . . 7 ⊢ ((𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥) ∧ ∪ (𝐵 ∩ 𝒫 𝑥) ⊆ (∪ 𝐵 ∩ ∪ 𝒫 𝑥)) → 𝑥 ⊆ (∪ 𝐵 ∩ ∪ 𝒫 𝑥))
11	9, 10	mpan2 704	. . . . . 6 ⊢ (𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥) → 𝑥 ⊆ (∪ 𝐵 ∩ ∪ 𝒫 𝑥))
12		ssin 4184	. . . . . 6 ⊢ ((𝑥 ⊆ ∪ 𝐵 ∧ 𝑥 ⊆ ∪ 𝒫 𝑥) ↔ 𝑥 ⊆ (∪ 𝐵 ∩ ∪ 𝒫 𝑥))
13	11, 12	sylibr 237	. . . . 5 ⊢ (𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥) → (𝑥 ⊆ ∪ 𝐵 ∧ 𝑥 ⊆ ∪ 𝒫 𝑥))
14	13	ss2abi 4014	. . . 4 ⊢ {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} ⊆ {𝑥 ∣ (𝑥 ⊆ ∪ 𝐵 ∧ 𝑥 ⊆ ∪ 𝒫 𝑥)}
15		ssexg 5284	. . . 4 ⊢ (({𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} ⊆ {𝑥 ∣ (𝑥 ⊆ ∪ 𝐵 ∧ 𝑥 ⊆ ∪ 𝒫 𝑥)} ∧ {𝑥 ∣ (𝑥 ⊆ ∪ 𝐵 ∧ 𝑥 ⊆ ∪ 𝒫 𝑥)} ∈ V) → {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} ∈ V)
16	14, 15	mpan 703	. . 3 ⊢ ({𝑥 ∣ (𝑥 ⊆ ∪ 𝐵 ∧ 𝑥 ⊆ ∪ 𝒫 𝑥)} ∈ V → {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} ∈ V)
17	7, 8, 16	3syl 19	. 2 ⊢ (𝐵 ∈ 𝑉 → {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} ∈ V)
18	1, 5, 6, 17	fvmptd3 7010	1 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2738  Vcvv 3450   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867  ‘cfv 6533  topGenctg 17522

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fv 6541  df-topgen 17528

This theorem is used by:  tgval2  23181  eltg  23182  tgdif0  23217