Theorem tgval 12876

Description: The topology generated by a basis. See also tgval2 14230 and tgval3 14237. (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		elex 2771	. 2 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V)
2		uniexg 4471	. . 3 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ∈ V)
3		abssexg 4212	. . 3 ⊢ (∪ 𝐵 ∈ V → {𝑥 ∣ (𝑥 ⊆ ∪ 𝐵 ∧ 𝑥 ⊆ ∪ 𝒫 𝑥)} ∈ V)
4		uniin 3856	. . . . . . 7 ⊢ ∪ (𝐵 ∩ 𝒫 𝑥) ⊆ (∪ 𝐵 ∩ ∪ 𝒫 𝑥)
5		sstr 3188	. . . . . . 7 ⊢ ((𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥) ∧ ∪ (𝐵 ∩ 𝒫 𝑥) ⊆ (∪ 𝐵 ∩ ∪ 𝒫 𝑥)) → 𝑥 ⊆ (∪ 𝐵 ∩ ∪ 𝒫 𝑥))
6	4, 5	mpan2 425	. . . . . 6 ⊢ (𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥) → 𝑥 ⊆ (∪ 𝐵 ∩ ∪ 𝒫 𝑥))
7		ssin 3382	. . . . . 6 ⊢ ((𝑥 ⊆ ∪ 𝐵 ∧ 𝑥 ⊆ ∪ 𝒫 𝑥) ↔ 𝑥 ⊆ (∪ 𝐵 ∩ ∪ 𝒫 𝑥))
8	6, 7	sylibr 134	. . . . 5 ⊢ (𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥) → (𝑥 ⊆ ∪ 𝐵 ∧ 𝑥 ⊆ ∪ 𝒫 𝑥))
9	8	ss2abi 3252	. . . 4 ⊢ {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} ⊆ {𝑥 ∣ (𝑥 ⊆ ∪ 𝐵 ∧ 𝑥 ⊆ ∪ 𝒫 𝑥)}
10		ssexg 4169	. . . 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)
11	9, 10	mpan 424	. . 3 ⊢ ({𝑥 ∣ (𝑥 ⊆ ∪ 𝐵 ∧ 𝑥 ⊆ ∪ 𝒫 𝑥)} ∈ V → {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} ∈ V)
12	2, 3, 11	3syl 17	. 2 ⊢ (𝐵 ∈ 𝑉 → {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} ∈ V)
13		ineq1 3354	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑥))
14	13	unieqd 3847	. . . . 5 ⊢ (𝑦 = 𝐵 → ∪ (𝑦 ∩ 𝒫 𝑥) = ∪ (𝐵 ∩ 𝒫 𝑥))
15	14	sseq2d 3210	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑥 ⊆ ∪ (𝑦 ∩ 𝒫 𝑥) ↔ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)))
16	15	abbidv 2311	. . 3 ⊢ (𝑦 = 𝐵 → {𝑥 ∣ 𝑥 ⊆ ∪ (𝑦 ∩ 𝒫 𝑥)} = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})
17		df-topgen 12874	. . 3 ⊢ topGen = (𝑦 ∈ V ↦ {𝑥 ∣ 𝑥 ⊆ ∪ (𝑦 ∩ 𝒫 𝑥)})
18	16, 17	fvmptg 5634	. 2 ⊢ ((𝐵 ∈ V ∧ {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} ∈ V) → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})
19	1, 12, 18	syl2anc 411	1 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  {cab 2179  Vcvv 2760   ∩ cin 3153   ⊆ wss 3154  𝒫 cpw 3602  ∪ cuni 3836  ‘cfv 5255  topGenctg 12868

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fv 5263  df-topgen 12874

This theorem is referenced by:  tgvalex  12877  tgval2  14230  eltg  14231