Theorem tgvalex 13560

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 13559	. 2 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑦 ∣ 𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦)})
2		inss1 3445	. . . . . . 7 ⊢ (𝐵 ∩ 𝒫 𝑦) ⊆ 𝐵
3	2	unissi 3942	. . . . . 6 ⊢ ∪ (𝐵 ∩ 𝒫 𝑦) ⊆ ∪ 𝐵
4		sstr 3250	. . . . . 6 ⊢ ((𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦) ∧ ∪ (𝐵 ∩ 𝒫 𝑦) ⊆ ∪ 𝐵) → 𝑦 ⊆ ∪ 𝐵)
5	3, 4	mpan2 425	. . . . 5 ⊢ (𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦) → 𝑦 ⊆ ∪ 𝐵)
6	5	ss2abi 3314	. . . 4 ⊢ {𝑦 ∣ 𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦)} ⊆ {𝑦 ∣ 𝑦 ⊆ ∪ 𝐵}
7		df-pw 3676	. . . 4 ⊢ 𝒫 ∪ 𝐵 = {𝑦 ∣ 𝑦 ⊆ ∪ 𝐵}
8	6, 7	sseqtrri 3277	. . 3 ⊢ {𝑦 ∣ 𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦)} ⊆ 𝒫 ∪ 𝐵
9		uniexg 4565	. . . 4 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ∈ V)
10	9	pwexd 4299	. . 3 ⊢ (𝐵 ∈ 𝑉 → 𝒫 ∪ 𝐵 ∈ V)
11		ssexg 4254	. . 3 ⊢ (({𝑦 ∣ 𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦)} ⊆ 𝒫 ∪ 𝐵 ∧ 𝒫 ∪ 𝐵 ∈ V) → {𝑦 ∣ 𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦)} ∈ V)
12	8, 10, 11	sylancr 414	. 2 ⊢ (𝐵 ∈ 𝑉 → {𝑦 ∣ 𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 𝑦)} ∈ V)
13	1, 12	eqeltrd 2311	1 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2205  {cab 2220  Vcvv 2815   ∩ cin 3213   ⊆ wss 3214  𝒫 cpw 3674  ∪ cuni 3919  ‘cfv 5357  topGenctg 13551

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-topgen 13557

This theorem is referenced by:  ptex  13561  mopnset  14826  tgcl  15055  tgidm  15065  tgss3  15069  2basgeng  15073  tgrest  15160  txvalex  15245  txval  15246  txbasval  15258