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Theorem istopon 22788
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 6929 . 2 (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ V)
2 uniexg 7737 . . . 4 (𝐽 ∈ Top → 𝐽 ∈ V)
3 eleq1 2816 . . . 4 (𝐵 = 𝐽 → (𝐵 ∈ V ↔ 𝐽 ∈ V))
42, 3syl5ibrcom 246 . . 3 (𝐽 ∈ Top → (𝐵 = 𝐽𝐵 ∈ V))
54imp 406 . 2 ((𝐽 ∈ Top ∧ 𝐵 = 𝐽) → 𝐵 ∈ V)
6 eqeq1 2731 . . . . . 6 (𝑏 = 𝐵 → (𝑏 = 𝑗𝐵 = 𝑗))
76rabbidv 3435 . . . . 5 (𝑏 = 𝐵 → {𝑗 ∈ Top ∣ 𝑏 = 𝑗} = {𝑗 ∈ Top ∣ 𝐵 = 𝑗})
8 df-topon 22787 . . . . 5 TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = 𝑗})
9 vpwex 5371 . . . . . . 7 𝒫 𝑏 ∈ V
109pwex 5374 . . . . . 6 𝒫 𝒫 𝑏 ∈ V
11 rabss 4065 . . . . . . 7 ({𝑗 ∈ Top ∣ 𝑏 = 𝑗} ⊆ 𝒫 𝒫 𝑏 ↔ ∀𝑗 ∈ Top (𝑏 = 𝑗𝑗 ∈ 𝒫 𝒫 𝑏))
12 pwuni 4943 . . . . . . . . . 10 𝑗 ⊆ 𝒫 𝑗
13 pweq 4612 . . . . . . . . . 10 (𝑏 = 𝑗 → 𝒫 𝑏 = 𝒫 𝑗)
1412, 13sseqtrrid 4031 . . . . . . . . 9 (𝑏 = 𝑗𝑗 ⊆ 𝒫 𝑏)
15 velpw 4603 . . . . . . . . 9 (𝑗 ∈ 𝒫 𝒫 𝑏𝑗 ⊆ 𝒫 𝑏)
1614, 15sylibr 233 . . . . . . . 8 (𝑏 = 𝑗𝑗 ∈ 𝒫 𝒫 𝑏)
1716a1i 11 . . . . . . 7 (𝑗 ∈ Top → (𝑏 = 𝑗𝑗 ∈ 𝒫 𝒫 𝑏))
1811, 17mprgbir 3063 . . . . . 6 {𝑗 ∈ Top ∣ 𝑏 = 𝑗} ⊆ 𝒫 𝒫 𝑏
1910, 18ssexi 5316 . . . . 5 {𝑗 ∈ Top ∣ 𝑏 = 𝑗} ∈ V
207, 8, 19fvmpt3i 7004 . . . 4 (𝐵 ∈ V → (TopOn‘𝐵) = {𝑗 ∈ Top ∣ 𝐵 = 𝑗})
2120eleq2d 2814 . . 3 (𝐵 ∈ V → (𝐽 ∈ (TopOn‘𝐵) ↔ 𝐽 ∈ {𝑗 ∈ Top ∣ 𝐵 = 𝑗}))
22 unieq 4914 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
2322eqeq2d 2738 . . . 4 (𝑗 = 𝐽 → (𝐵 = 𝑗𝐵 = 𝐽))
2423elrab 3680 . . 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 205  wa 395   = wceq 1534  wcel 2099  {crab 3427  Vcvv 3469  wss 3944  𝒫 cpw 4598   cuni 4903  cfv 6542  Topctop 22769  TopOnctopon 22786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-topon 22787
This theorem is referenced by:  topontop  22789  toponuni  22790  toptopon  22793  toponcom  22804  istps2  22811  tgtopon  22848  distopon  22874  indistopon  22878  fctop  22881  cctop  22883  ppttop  22884  epttop  22886  mretopd  22970  toponmre  22971  resttopon  23039  resttopon2  23046  kgentopon  23416  txtopon  23469  pttopon  23474  xkotopon  23478  qtoptopon  23582  flimtopon  23848  fclstopon  23890  fclsfnflim  23905  utoptopon  24115  qtopt1  33359  neibastop1  35766  onsuctopon  35841  rfcnpre1  44294  cnfex  44303  icccncfext  45188  stoweidlem47  45348
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