Theorem tgvalex 13617

Description: The topology generated by a basis is a set. (Contributed by Jim Kingdon, 4-Mar-2023.)

Assertion

Ref	Expression
tgvalex	⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) ∈ V)

Proof of Theorem tgvalex

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tgval 13616	. 2 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑦 ∣ 𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦)})
2		inss1 3451	. . . . . . 7 ⊢ (𝐵 ∩ 𝒫 𝑦) ⊆ 𝐵
3	2	unissi 3958	. . . . . 6 ⊢ ∪ (𝐵 ∩ 𝒫 𝑦) ⊆ ∪ 𝐵
4		sstr 3256	. . . . . 6 ⊢ ((𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦) ∧ ∪ (𝐵 ∩ 𝒫 𝑦) ⊆ ∪ 𝐵) → 𝑦 ⊆ ∪ 𝐵)
5	3, 4	mpan2 429	. . . . 5 ⊢ (𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦) → 𝑦 ⊆ ∪ 𝐵)
6	5	ss2abi 3320	. . . 4 ⊢ {𝑦 ∣ 𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦)} ⊆ {𝑦 ∣ 𝑦 ⊆ ∪ 𝐵}
7		df-pw 3690	. . . 4 ⊢ 𝒫 ∪ 𝐵 = {𝑦 ∣ 𝑦 ⊆ ∪ 𝐵}
8	6, 7	sseqtrri 3283	. . 3 ⊢ {𝑦 ∣ 𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦)} ⊆ 𝒫 ∪ 𝐵
9		uniexg 4585	. . . 4 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ∈ V)
10	9	pwexd 4318	. . 3 ⊢ (𝐵 ∈ 𝑉 → 𝒫 ∪ 𝐵 ∈ V)
11		ssexg 4272	. . 3 ⊢ (({𝑦 ∣ 𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦)} ⊆ 𝒫 ∪ 𝐵 ∧ 𝒫 ∪ 𝐵 ∈ V) → {𝑦 ∣ 𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦)} ∈ V)
12	8, 10, 11	sylancr 418	. 2 ⊢ (𝐵 ∈ 𝑉 → {𝑦 ∣ 𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦)} ∈ V)
13	1, 12	eqeltrd 2315	1 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2209  {cab 2224  Vcvv 2821   ∩ cin 3219   ⊆ wss 3220  𝒫 cpw 3688  ∪ cuni 3935  ‘cfv 5377  topGenctg 13608

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-topgen 13614

This theorem is used by:  ptex  13618  mopnset  14889  tgcl  15165  tgidm  15175  tgss3  15179  2basgeng  15183  tgrest  15270  txvalex  15355  txval  15356  txbasval  15368