Theorem tgval 12689

Description: The topology generated by a basis. See also tgval2 12691 and tgval3 12698. (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 2737	. 2 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V)
2		uniexg 4417	. . 3 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ∈ V)
3		abssexg 4161	. . 3 ⊢ (∪ 𝐵 ∈ V → {𝑥 ∣ (𝑥 ⊆ ∪ 𝐵 ∧ 𝑥 ⊆ ∪ 𝒫 𝑥)} ∈ V)
4		uniin 3809	. . . . . . 7 ⊢ ∪ (𝐵 ∩ 𝒫 𝑥) ⊆ (∪ 𝐵 ∩ ∪ 𝒫 𝑥)
5		sstr 3150	. . . . . . 7 ⊢ ((𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥) ∧ ∪ (𝐵 ∩ 𝒫 𝑥) ⊆ (∪ 𝐵 ∩ ∪ 𝒫 𝑥)) → 𝑥 ⊆ (∪ 𝐵 ∩ ∪ 𝒫 𝑥))
6	4, 5	mpan2 422	. . . . . 6 ⊢ (𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥) → 𝑥 ⊆ (∪ 𝐵 ∩ ∪ 𝒫 𝑥))
7		ssin 3344	. . . . . 6 ⊢ ((𝑥 ⊆ ∪ 𝐵 ∧ 𝑥 ⊆ ∪ 𝒫 𝑥) ↔ 𝑥 ⊆ (∪ 𝐵 ∩ ∪ 𝒫 𝑥))
8	6, 7	sylibr 133	. . . . 5 ⊢ (𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥) → (𝑥 ⊆ ∪ 𝐵 ∧ 𝑥 ⊆ ∪ 𝒫 𝑥))
9	8	ss2abi 3214	. . . 4 ⊢ {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} ⊆ {𝑥 ∣ (𝑥 ⊆ ∪ 𝐵 ∧ 𝑥 ⊆ ∪ 𝒫 𝑥)}
10		ssexg 4121	. . . 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 421	. . 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 3316	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑥))
14	13	unieqd 3800	. . . . 5 ⊢ (𝑦 = 𝐵 → ∪ (𝑦 ∩ 𝒫 𝑥) = ∪ (𝐵 ∩ 𝒫 𝑥))
15	14	sseq2d 3172	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑥 ⊆ ∪ (𝑦 ∩ 𝒫 𝑥) ↔ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)))
16	15	abbidv 2284	. . 3 ⊢ (𝑦 = 𝐵 → {𝑥 ∣ 𝑥 ⊆ ∪ (𝑦 ∩ 𝒫 𝑥)} = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})
17		df-topgen 12577	. . 3 ⊢ topGen = (𝑦 ∈ V ↦ {𝑥 ∣ 𝑥 ⊆ ∪ (𝑦 ∩ 𝒫 𝑥)})
18	16, 17	fvmptg 5562	. 2 ⊢ ((𝐵 ∈ V ∧ {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} ∈ V) → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})
19	1, 12, 18	syl2anc 409	1 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343   ∈ wcel 2136  {cab 2151  Vcvv 2726   ∩ cin 3115   ⊆ wss 3116  𝒫 cpw 3559  ∪ cuni 3789  ‘cfv 5188  topGenctg 12571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-topgen 12577

This theorem is referenced by:  tgvalex  12690  tgval2  12691  eltg  12692