Theorem tgval 23280

Description: The topology generated by a basis. See also tgval2 23281 and tgval3 23288. (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 17621	. 2 ⊢ topGen = (𝑦 ∈ V ↦ {𝑥 ∣ 𝑥 ⊆ ∪ (𝑦 ∩ 𝒫 𝑥)})
2		ineq1 4109	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑥))
3	2	unieqd 4879	. . . 4 ⊢ (𝑦 = 𝐵 → ∪ (𝑦 ∩ 𝒫 𝑥) = ∪ (𝐵 ∩ 𝒫 𝑥))
4	3	sseq2d 3941	. . 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 7759	. . 3 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ∈ V)
8		abssexg 5343	. . 3 ⊢ (∪ 𝐵 ∈ V → {𝑥 ∣ (𝑥 ⊆ ∪ 𝐵 ∧ 𝑥 ⊆ ∪ 𝒫 𝑥)} ∈ V)
9		uniin 4890	. . . . . . 7 ⊢ ∪ (𝐵 ∩ 𝒫 𝑥) ⊆ (∪ 𝐵 ∩ ∪ 𝒫 𝑥)
10		sstr 3917	. . . . . . 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 4134	. . . . . 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 3990	. . . 4 ⊢ {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} ⊆ {𝑥 ∣ (𝑥 ⊆ ∪ 𝐵 ∧ 𝑥 ⊆ ∪ 𝒫 𝑥)}
15		ssexg 5280	. . . 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 7020	1 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738  Vcvv 3450   ⊆ wss 3896   ∩ cin 4088  𝒫 cpw 4556  ∪ cuni 4866  ‘cfv 6541  topGenctg 17615

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pow 5326  ax-pr 5390  ax-un 7753

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6496  df-fun 6543  df-fv 6549  df-topgen 17621

This theorem is used by:  tgval2  23281  eltg  23282  tgdif0  23317