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Theorem istopon 20765
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 6259 . 2 (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ V)
2 uniexg 6997 . . . 4 (𝐽 ∈ Top → 𝐽 ∈ V)
3 eleq1 2718 . . . 4 (𝐵 = 𝐽 → (𝐵 ∈ V ↔ 𝐽 ∈ V))
42, 3syl5ibrcom 237 . . 3 (𝐽 ∈ Top → (𝐵 = 𝐽𝐵 ∈ V))
54imp 444 . 2 ((𝐽 ∈ Top ∧ 𝐵 = 𝐽) → 𝐵 ∈ V)
6 eqeq1 2655 . . . . . 6 (𝑏 = 𝐵 → (𝑏 = 𝑗𝐵 = 𝑗))
76rabbidv 3220 . . . . 5 (𝑏 = 𝐵 → {𝑗 ∈ Top ∣ 𝑏 = 𝑗} = {𝑗 ∈ Top ∣ 𝐵 = 𝑗})
8 df-topon 20764 . . . . 5 TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = 𝑗})
9 vpwex 4879 . . . . . . 7 𝒫 𝑏 ∈ V
109pwex 4878 . . . . . 6 𝒫 𝒫 𝑏 ∈ V
11 rabss 3712 . . . . . . 7 ({𝑗 ∈ Top ∣ 𝑏 = 𝑗} ⊆ 𝒫 𝒫 𝑏 ↔ ∀𝑗 ∈ Top (𝑏 = 𝑗𝑗 ∈ 𝒫 𝒫 𝑏))
12 pwuni 4506 . . . . . . . . . 10 𝑗 ⊆ 𝒫 𝑗
13 pweq 4194 . . . . . . . . . 10 (𝑏 = 𝑗 → 𝒫 𝑏 = 𝒫 𝑗)
1412, 13syl5sseqr 3687 . . . . . . . . 9 (𝑏 = 𝑗𝑗 ⊆ 𝒫 𝑏)
15 selpw 4198 . . . . . . . . 9 (𝑗 ∈ 𝒫 𝒫 𝑏𝑗 ⊆ 𝒫 𝑏)
1614, 15sylibr 224 . . . . . . . 8 (𝑏 = 𝑗𝑗 ∈ 𝒫 𝒫 𝑏)
1716a1i 11 . . . . . . 7 (𝑗 ∈ Top → (𝑏 = 𝑗𝑗 ∈ 𝒫 𝒫 𝑏))
1811, 17mprgbir 2956 . . . . . 6 {𝑗 ∈ Top ∣ 𝑏 = 𝑗} ⊆ 𝒫 𝒫 𝑏
1910, 18ssexi 4836 . . . . 5 {𝑗 ∈ Top ∣ 𝑏 = 𝑗} ∈ V
207, 8, 19fvmpt3i 6326 . . . 4 (𝐵 ∈ V → (TopOn‘𝐵) = {𝑗 ∈ Top ∣ 𝐵 = 𝑗})
2120eleq2d 2716 . . 3 (𝐵 ∈ V → (𝐽 ∈ (TopOn‘𝐵) ↔ 𝐽 ∈ {𝑗 ∈ Top ∣ 𝐵 = 𝑗}))
22 unieq 4476 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
2322eqeq2d 2661 . . . 4 (𝑗 = 𝐽 → (𝐵 = 𝑗𝐵 = 𝐽))
2423elrab 3396 . . 3 (𝐽 ∈ {𝑗 ∈ Top ∣ 𝐵 = 𝑗} ↔ (𝐽 ∈ Top ∧ 𝐵 = 𝐽))
2521, 24syl6bb 276 . 2 (𝐵 ∈ V → (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = 𝐽)))
261, 5, 25pm5.21nii 367 1 (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  {crab 2945  Vcvv 3231  wss 3607  𝒫 cpw 4191   cuni 4468  cfv 5926  Topctop 20746  TopOnctopon 20763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-topon 20764
This theorem is referenced by:  topontop  20766  toponuni  20767  toptopon  20770  toponcom  20780  istps2  20787  tgtopon  20823  distopon  20849  indistopon  20853  fctop  20856  cctop  20858  ppttop  20859  epttop  20861  mretopd  20944  toponmre  20945  resttopon  21013  resttopon2  21020  kgentopon  21389  txtopon  21442  pttopon  21447  xkotopon  21451  qtoptopon  21555  flimtopon  21821  fclstopon  21863  fclsfnflim  21878  utoptopon  22087  qtopt1  30030  neibastop1  32479  onsuctopon  32558  rfcnpre1  39492  cnfex  39501  icccncfext  40418  stoweidlem47  40582
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