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Theorem istopon 22918
Description: Property of being a topology with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 13-Aug-2015.)
Assertion
Ref Expression
istopon (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = 𝐽))

Proof of Theorem istopon
Dummy variables 𝑏 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 6944 . 2 (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ V)
2 uniexg 7760 . . . 4 (𝐽 ∈ Top → 𝐽 ∈ V)
3 eleq1 2829 . . . 4 (𝐵 = 𝐽 → (𝐵 ∈ V ↔ 𝐽 ∈ V))
42, 3syl5ibrcom 247 . . 3 (𝐽 ∈ Top → (𝐵 = 𝐽𝐵 ∈ V))
54imp 406 . 2 ((𝐽 ∈ Top ∧ 𝐵 = 𝐽) → 𝐵 ∈ V)
6 eqeq1 2741 . . . . . 6 (𝑏 = 𝐵 → (𝑏 = 𝑗𝐵 = 𝑗))
76rabbidv 3444 . . . . 5 (𝑏 = 𝐵 → {𝑗 ∈ Top ∣ 𝑏 = 𝑗} = {𝑗 ∈ Top ∣ 𝐵 = 𝑗})
8 df-topon 22917 . . . . 5 TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = 𝑗})
9 vpwex 5377 . . . . . . 7 𝒫 𝑏 ∈ V
109pwex 5380 . . . . . 6 𝒫 𝒫 𝑏 ∈ V
11 rabss 4072 . . . . . . 7 ({𝑗 ∈ Top ∣ 𝑏 = 𝑗} ⊆ 𝒫 𝒫 𝑏 ↔ ∀𝑗 ∈ Top (𝑏 = 𝑗𝑗 ∈ 𝒫 𝒫 𝑏))
12 pwuni 4945 . . . . . . . . . 10 𝑗 ⊆ 𝒫 𝑗
13 pweq 4614 . . . . . . . . . 10 (𝑏 = 𝑗 → 𝒫 𝑏 = 𝒫 𝑗)
1412, 13sseqtrrid 4027 . . . . . . . . 9 (𝑏 = 𝑗𝑗 ⊆ 𝒫 𝑏)
15 velpw 4605 . . . . . . . . 9 (𝑗 ∈ 𝒫 𝒫 𝑏𝑗 ⊆ 𝒫 𝑏)
1614, 15sylibr 234 . . . . . . . 8 (𝑏 = 𝑗𝑗 ∈ 𝒫 𝒫 𝑏)
1716a1i 11 . . . . . . 7 (𝑗 ∈ Top → (𝑏 = 𝑗𝑗 ∈ 𝒫 𝒫 𝑏))
1811, 17mprgbir 3068 . . . . . 6 {𝑗 ∈ Top ∣ 𝑏 = 𝑗} ⊆ 𝒫 𝒫 𝑏
1910, 18ssexi 5322 . . . . 5 {𝑗 ∈ Top ∣ 𝑏 = 𝑗} ∈ V
207, 8, 19fvmpt3i 7021 . . . 4 (𝐵 ∈ V → (TopOn‘𝐵) = {𝑗 ∈ Top ∣ 𝐵 = 𝑗})
2120eleq2d 2827 . . 3 (𝐵 ∈ V → (𝐽 ∈ (TopOn‘𝐵) ↔ 𝐽 ∈ {𝑗 ∈ Top ∣ 𝐵 = 𝑗}))
22 unieq 4918 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
2322eqeq2d 2748 . . . 4 (𝑗 = 𝐽 → (𝐵 = 𝑗𝐵 = 𝐽))
2423elrab 3692 . . 3 (𝐽 ∈ {𝑗 ∈ Top ∣ 𝐵 = 𝑗} ↔ (𝐽 ∈ Top ∧ 𝐵 = 𝐽))
2521, 24bitrdi 287 . 2 (𝐵 ∈ V → (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = 𝐽)))
261, 5, 25pm5.21nii 378 1 (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  wss 3951  𝒫 cpw 4600   cuni 4907  cfv 6561  Topctop 22899  TopOnctopon 22916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-topon 22917
This theorem is referenced by:  topontop  22919  toponuni  22920  toptopon  22923  toponcom  22934  istps2  22941  tgtopon  22978  distopon  23004  indistopon  23008  fctop  23011  cctop  23013  ppttop  23014  epttop  23016  mretopd  23100  toponmre  23101  resttopon  23169  resttopon2  23176  kgentopon  23546  txtopon  23599  pttopon  23604  xkotopon  23608  qtoptopon  23712  flimtopon  23978  fclstopon  24020  fclsfnflim  24035  utoptopon  24245  qtopt1  33834  neibastop1  36360  onsuctopon  36435  rfcnpre1  45024  cnfex  45033  icccncfext  45902  stoweidlem47  46062
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