ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tgval GIF version

Theorem tgval 11917
Description: The topology generated by a basis. See also tgval2 11919 and tgval3 11926. (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 elex 2644 . 2 (𝐵𝑉𝐵 ∈ V)
2 uniexg 4290 . . 3 (𝐵𝑉 𝐵 ∈ V)
3 abssexg 4038 . . 3 ( 𝐵 ∈ V → {𝑥 ∣ (𝑥 𝐵𝑥 𝒫 𝑥)} ∈ V)
4 uniin 3695 . . . . . . 7 (𝐵 ∩ 𝒫 𝑥) ⊆ ( 𝐵 𝒫 𝑥)
5 sstr 3047 . . . . . . 7 ((𝑥 (𝐵 ∩ 𝒫 𝑥) ∧ (𝐵 ∩ 𝒫 𝑥) ⊆ ( 𝐵 𝒫 𝑥)) → 𝑥 ⊆ ( 𝐵 𝒫 𝑥))
64, 5mpan2 417 . . . . . 6 (𝑥 (𝐵 ∩ 𝒫 𝑥) → 𝑥 ⊆ ( 𝐵 𝒫 𝑥))
7 ssin 3237 . . . . . 6 ((𝑥 𝐵𝑥 𝒫 𝑥) ↔ 𝑥 ⊆ ( 𝐵 𝒫 𝑥))
86, 7sylibr 133 . . . . 5 (𝑥 (𝐵 ∩ 𝒫 𝑥) → (𝑥 𝐵𝑥 𝒫 𝑥))
98ss2abi 3108 . . . 4 {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ⊆ {𝑥 ∣ (𝑥 𝐵𝑥 𝒫 𝑥)}
10 ssexg 3999 . . . 4 (({𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ⊆ {𝑥 ∣ (𝑥 𝐵𝑥 𝒫 𝑥)} ∧ {𝑥 ∣ (𝑥 𝐵𝑥 𝒫 𝑥)} ∈ V) → {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ∈ V)
119, 10mpan 416 . . 3 ({𝑥 ∣ (𝑥 𝐵𝑥 𝒫 𝑥)} ∈ V → {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ∈ V)
122, 3, 113syl 17 . 2 (𝐵𝑉 → {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ∈ V)
13 ineq1 3209 . . . . . 6 (𝑦 = 𝐵 → (𝑦 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑥))
1413unieqd 3686 . . . . 5 (𝑦 = 𝐵 (𝑦 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑥))
1514sseq2d 3069 . . . 4 (𝑦 = 𝐵 → (𝑥 (𝑦 ∩ 𝒫 𝑥) ↔ 𝑥 (𝐵 ∩ 𝒫 𝑥)))
1615abbidv 2212 . . 3 (𝑦 = 𝐵 → {𝑥𝑥 (𝑦 ∩ 𝒫 𝑥)} = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})
17 df-topgen 11841 . . 3 topGen = (𝑦 ∈ V ↦ {𝑥𝑥 (𝑦 ∩ 𝒫 𝑥)})
1816, 17fvmptg 5415 . 2 ((𝐵 ∈ V ∧ {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ∈ V) → (topGen‘𝐵) = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})
191, 12, 18syl2anc 404 1 (𝐵𝑉 → (topGen‘𝐵) = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1296  wcel 1445  {cab 2081  Vcvv 2633  cin 3012  wss 3013  𝒫 cpw 3449   cuni 3675  cfv 5049  topGenctg 11835
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-iota 5014  df-fun 5051  df-fv 5057  df-topgen 11841
This theorem is referenced by:  tgvalex  11918  tgval2  11919  eltg  11920
  Copyright terms: Public domain W3C validator