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Theorem tgval 22977
Description: The topology generated by a basis. See also tgval2 22978 and tgval3 22985. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
Assertion
Ref Expression
tgval (𝐵𝑉 → (topGen‘𝐵) = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})
Distinct variable groups:   𝑥,𝐵   𝑥,𝑉

Proof of Theorem tgval
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-topgen 17489 . 2 topGen = (𝑦 ∈ V ↦ {𝑥𝑥 (𝑦 ∩ 𝒫 𝑥)})
2 ineq1 4220 . . . . 5 (𝑦 = 𝐵 → (𝑦 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑥))
32unieqd 4924 . . . 4 (𝑦 = 𝐵 (𝑦 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑥))
43sseq2d 4027 . . 3 (𝑦 = 𝐵 → (𝑥 (𝑦 ∩ 𝒫 𝑥) ↔ 𝑥 (𝐵 ∩ 𝒫 𝑥)))
54abbidv 2805 . 2 (𝑦 = 𝐵 → {𝑥𝑥 (𝑦 ∩ 𝒫 𝑥)} = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})
6 elex 3498 . 2 (𝐵𝑉𝐵 ∈ V)
7 uniexg 7758 . . 3 (𝐵𝑉 𝐵 ∈ V)
8 abssexg 5387 . . 3 ( 𝐵 ∈ V → {𝑥 ∣ (𝑥 𝐵𝑥 𝒫 𝑥)} ∈ V)
9 uniin 4935 . . . . . . 7 (𝐵 ∩ 𝒫 𝑥) ⊆ ( 𝐵 𝒫 𝑥)
10 sstr 4003 . . . . . . 7 ((𝑥 (𝐵 ∩ 𝒫 𝑥) ∧ (𝐵 ∩ 𝒫 𝑥) ⊆ ( 𝐵 𝒫 𝑥)) → 𝑥 ⊆ ( 𝐵 𝒫 𝑥))
119, 10mpan2 691 . . . . . 6 (𝑥 (𝐵 ∩ 𝒫 𝑥) → 𝑥 ⊆ ( 𝐵 𝒫 𝑥))
12 ssin 4246 . . . . . 6 ((𝑥 𝐵𝑥 𝒫 𝑥) ↔ 𝑥 ⊆ ( 𝐵 𝒫 𝑥))
1311, 12sylibr 234 . . . . 5 (𝑥 (𝐵 ∩ 𝒫 𝑥) → (𝑥 𝐵𝑥 𝒫 𝑥))
1413ss2abi 4076 . . . 4 {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ⊆ {𝑥 ∣ (𝑥 𝐵𝑥 𝒫 𝑥)}
15 ssexg 5328 . . . 4 (({𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ⊆ {𝑥 ∣ (𝑥 𝐵𝑥 𝒫 𝑥)} ∧ {𝑥 ∣ (𝑥 𝐵𝑥 𝒫 𝑥)} ∈ V) → {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ∈ V)
1614, 15mpan 690 . . 3 ({𝑥 ∣ (𝑥 𝐵𝑥 𝒫 𝑥)} ∈ V → {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ∈ V)
177, 8, 163syl 18 . 2 (𝐵𝑉 → {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ∈ V)
181, 5, 6, 17fvmptd3 7038 1 (𝐵𝑉 → (topGen‘𝐵) = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  {cab 2711  Vcvv 3477  cin 3961  wss 3962  𝒫 cpw 4604   cuni 4911  cfv 6562  topGenctg 17483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-topgen 17489
This theorem is referenced by:  tgval2  22978  eltg  22979  tgdif0  23014
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