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Theorem tgvalex 13095
Description: The topology generated by a basis is a set. (Contributed by Jim Kingdon, 4-Mar-2023.)
Assertion
Ref Expression
tgvalex (𝐵𝑉 → (topGen‘𝐵) ∈ V)

Proof of Theorem tgvalex
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 tgval 13094 . 2 (𝐵𝑉 → (topGen‘𝐵) = {𝑦𝑦 (𝐵 ∩ 𝒫 𝑦)})
2 inss1 3393 . . . . . . 7 (𝐵 ∩ 𝒫 𝑦) ⊆ 𝐵
32unissi 3873 . . . . . 6 (𝐵 ∩ 𝒫 𝑦) ⊆ 𝐵
4 sstr 3201 . . . . . 6 ((𝑦 (𝐵 ∩ 𝒫 𝑦) ∧ (𝐵 ∩ 𝒫 𝑦) ⊆ 𝐵) → 𝑦 𝐵)
53, 4mpan2 425 . . . . 5 (𝑦 (𝐵 ∩ 𝒫 𝑦) → 𝑦 𝐵)
65ss2abi 3265 . . . 4 {𝑦𝑦 (𝐵 ∩ 𝒫 𝑦)} ⊆ {𝑦𝑦 𝐵}
7 df-pw 3618 . . . 4 𝒫 𝐵 = {𝑦𝑦 𝐵}
86, 7sseqtrri 3228 . . 3 {𝑦𝑦 (𝐵 ∩ 𝒫 𝑦)} ⊆ 𝒫 𝐵
9 uniexg 4486 . . . 4 (𝐵𝑉 𝐵 ∈ V)
109pwexd 4225 . . 3 (𝐵𝑉 → 𝒫 𝐵 ∈ V)
11 ssexg 4183 . . 3 (({𝑦𝑦 (𝐵 ∩ 𝒫 𝑦)} ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ∈ V) → {𝑦𝑦 (𝐵 ∩ 𝒫 𝑦)} ∈ V)
128, 10, 11sylancr 414 . 2 (𝐵𝑉 → {𝑦𝑦 (𝐵 ∩ 𝒫 𝑦)} ∈ V)
131, 12eqeltrd 2282 1 (𝐵𝑉 → (topGen‘𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2176  {cab 2191  Vcvv 2772  cin 3165  wss 3166  𝒫 cpw 3616   cuni 3850  cfv 5271  topGenctg 13086
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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480
This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-topgen 13092
This theorem is referenced by:  ptex  13096  mopnset  14314  tgcl  14536  tgidm  14546  tgss3  14550  2basgeng  14554  tgrest  14641  txvalex  14726  txval  14727  txbasval  14739
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