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Theorem dmtopon 22929
Description: The domain of TopOn is the universal class V. (Contributed by BJ, 29-Apr-2021.)
Assertion
Ref Expression
dmtopon dom TopOn = V

Proof of Theorem dmtopon
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vpwex 5377 . . . 4 𝒫 𝑥 ∈ V
21pwex 5380 . . 3 𝒫 𝒫 𝑥 ∈ V
3 eqcom 2744 . . . . 5 (𝑥 = 𝑦 𝑦 = 𝑥)
43rabbii 3442 . . . 4 {𝑦 ∈ Top ∣ 𝑥 = 𝑦} = {𝑦 ∈ Top ∣ 𝑦 = 𝑥}
5 rabssab 4085 . . . . 5 {𝑦 ∈ Top ∣ 𝑦 = 𝑥} ⊆ {𝑦 𝑦 = 𝑥}
6 pwpwssunieq 5104 . . . . 5 {𝑦 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥
75, 6sstri 3993 . . . 4 {𝑦 ∈ Top ∣ 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥
84, 7eqsstri 4030 . . 3 {𝑦 ∈ Top ∣ 𝑥 = 𝑦} ⊆ 𝒫 𝒫 𝑥
92, 8ssexi 5322 . 2 {𝑦 ∈ Top ∣ 𝑥 = 𝑦} ∈ V
10 df-topon 22917 . 2 TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = 𝑦})
119, 10dmmpti 6712 1 dom TopOn = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  {cab 2714  {crab 3436  Vcvv 3480  𝒫 cpw 4600   cuni 4907  dom cdm 5685  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
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-fun 6563  df-fn 6564  df-topon 22917
This theorem is referenced by:  fntopon  22930
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