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Theorem istopon 22933
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 7758 . . . 4 (𝐽 ∈ Top → 𝐽 ∈ V)
3 eleq1 2826 . . . 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 2738 . . . . . 6 (𝑏 = 𝐵 → (𝑏 = 𝑗𝐵 = 𝑗))
76rabbidv 3440 . . . . 5 (𝑏 = 𝐵 → {𝑗 ∈ Top ∣ 𝑏 = 𝑗} = {𝑗 ∈ Top ∣ 𝐵 = 𝑗})
8 df-topon 22932 . . . . 5 TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = 𝑗})
9 vpwex 5382 . . . . . . 7 𝒫 𝑏 ∈ V
109pwex 5385 . . . . . 6 𝒫 𝒫 𝑏 ∈ V
11 rabss 4081 . . . . . . 7 ({𝑗 ∈ Top ∣ 𝑏 = 𝑗} ⊆ 𝒫 𝒫 𝑏 ↔ ∀𝑗 ∈ Top (𝑏 = 𝑗𝑗 ∈ 𝒫 𝒫 𝑏))
12 pwuni 4949 . . . . . . . . . 10 𝑗 ⊆ 𝒫 𝑗
13 pweq 4618 . . . . . . . . . 10 (𝑏 = 𝑗 → 𝒫 𝑏 = 𝒫 𝑗)
1412, 13sseqtrrid 4048 . . . . . . . . 9 (𝑏 = 𝑗𝑗 ⊆ 𝒫 𝑏)
15 velpw 4609 . . . . . . . . 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 3065 . . . . . 6 {𝑗 ∈ Top ∣ 𝑏 = 𝑗} ⊆ 𝒫 𝒫 𝑏
1910, 18ssexi 5327 . . . . 5 {𝑗 ∈ Top ∣ 𝑏 = 𝑗} ∈ V
207, 8, 19fvmpt3i 7020 . . . 4 (𝐵 ∈ V → (TopOn‘𝐵) = {𝑗 ∈ Top ∣ 𝐵 = 𝑗})
2120eleq2d 2824 . . 3 (𝐵 ∈ V → (𝐽 ∈ (TopOn‘𝐵) ↔ 𝐽 ∈ {𝑗 ∈ Top ∣ 𝐵 = 𝑗}))
22 unieq 4922 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
2322eqeq2d 2745 . . . 4 (𝑗 = 𝐽 → (𝐵 = 𝑗𝐵 = 𝐽))
2423elrab 3694 . . 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 1536  wcel 2105  {crab 3432  Vcvv 3477  wss 3962  𝒫 cpw 4604   cuni 4911  cfv 6562  Topctop 22914  TopOnctopon 22931
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-topon 22932
This theorem is referenced by:  topontop  22934  toponuni  22935  toptopon  22938  toponcom  22949  istps2  22956  tgtopon  22993  distopon  23019  indistopon  23023  fctop  23026  cctop  23028  ppttop  23029  epttop  23031  mretopd  23115  toponmre  23116  resttopon  23184  resttopon2  23191  kgentopon  23561  txtopon  23614  pttopon  23619  xkotopon  23623  qtoptopon  23727  flimtopon  23993  fclstopon  24035  fclsfnflim  24050  utoptopon  24260  qtopt1  33795  neibastop1  36341  onsuctopon  36416  rfcnpre1  44956  cnfex  44965  icccncfext  45842  stoweidlem47  46002
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