Theorem tgval 22984

Description: The topology generated by a basis. See also tgval2 22985 and tgval3 22992. (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 17444	. 2 ⊢ topGen = (𝑦 ∈ V ↦ {𝑥 ∣ 𝑥 ⊆ ∪ (𝑦 ∩ 𝒫 𝑥)})
2		ineq1 4156	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑥))
3	2	unieqd 4868	. . . 4 ⊢ (𝑦 = 𝐵 → ∪ (𝑦 ∩ 𝒫 𝑥) = ∪ (𝐵 ∩ 𝒫 𝑥))
4	3	sseq2d 3959	. . 3 ⊢ (𝑦 = 𝐵 → (𝑥 ⊆ ∪ (𝑦 ∩ 𝒫 𝑥) ↔ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)))
5	4	abbidv 2818	. 2 ⊢ (𝑦 = 𝐵 → {𝑥 ∣ 𝑥 ⊆ ∪ (𝑦 ∩ 𝒫 𝑥)} = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})
6		elex 3465	. 2 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V)
7		uniexg 7708	. . 3 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ∈ V)
8		abssexg 5329	. . 3 ⊢ (∪ 𝐵 ∈ V → {𝑥 ∣ (𝑥 ⊆ ∪ 𝐵 ∧ 𝑥 ⊆ ∪ 𝒫 𝑥)} ∈ V)
9		uniin 4879	. . . . . . 7 ⊢ ∪ (𝐵 ∩ 𝒫 𝑥) ⊆ (∪ 𝐵 ∩ ∪ 𝒫 𝑥)
10		sstr 3935	. . . . . . 7 ⊢ ((𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥) ∧ ∪ (𝐵 ∩ 𝒫 𝑥) ⊆ (∪ 𝐵 ∩ ∪ 𝒫 𝑥)) → 𝑥 ⊆ (∪ 𝐵 ∩ ∪ 𝒫 𝑥))
11	9, 10	mpan2 699	. . . . . 6 ⊢ (𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥) → 𝑥 ⊆ (∪ 𝐵 ∩ ∪ 𝒫 𝑥))
12		ssin 4181	. . . . . 6 ⊢ ((𝑥 ⊆ ∪ 𝐵 ∧ 𝑥 ⊆ ∪ 𝒫 𝑥) ↔ 𝑥 ⊆ (∪ 𝐵 ∩ ∪ 𝒫 𝑥))
13	11, 12	sylibr 236	. . . . 5 ⊢ (𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥) → (𝑥 ⊆ ∪ 𝐵 ∧ 𝑥 ⊆ ∪ 𝒫 𝑥))
14	13	ss2abi 4010	. . . 4 ⊢ {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} ⊆ {𝑥 ∣ (𝑥 ⊆ ∪ 𝐵 ∧ 𝑥 ⊆ ∪ 𝒫 𝑥)}
15		ssexg 5269	. . . 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 698	. . 3 ⊢ ({𝑥 ∣ (𝑥 ⊆ ∪ 𝐵 ∧ 𝑥 ⊆ ∪ 𝒫 𝑥)} ∈ V → {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} ∈ V)
17	7, 8, 16	3syl 18	. 2 ⊢ (𝐵 ∈ 𝑉 → {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} ∈ V)
18	1, 5, 6, 17	fvmptd3 6984	1 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1550   ∈ wcel 2132  {cab 2730  Vcvv 3444   ∩ cin 3894   ⊆ wss 3895  𝒫 cpw 4545  ∪ cuni 4855  ‘cfv 6506  topGenctg 17438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-sep 5236  ax-pow 5312  ax-pr 5380  ax-un 7703

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ral 3067  df-rex 3077  df-rab 3405  df-v 3446  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-br 5091  df-opab 5153  df-mpt 5172  df-id 5531  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-iota 6462  df-fun 6508  df-fv 6514  df-topgen 17444

This theorem is referenced by:  tgval2  22985  eltg  22986  tgdif0  23021