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Theorem dmtopon 21820
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 5270 . . . 4 𝒫 𝑥 ∈ V
21pwex 5273 . . 3 𝒫 𝒫 𝑥 ∈ V
3 eqcom 2744 . . . . 5 (𝑥 = 𝑦 𝑦 = 𝑥)
43rabbii 3383 . . . 4 {𝑦 ∈ Top ∣ 𝑥 = 𝑦} = {𝑦 ∈ Top ∣ 𝑦 = 𝑥}
5 rabssab 3998 . . . . 5 {𝑦 ∈ Top ∣ 𝑦 = 𝑥} ⊆ {𝑦 𝑦 = 𝑥}
6 pwpwssunieq 5012 . . . . 5 {𝑦 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥
75, 6sstri 3910 . . . 4 {𝑦 ∈ Top ∣ 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥
84, 7eqsstri 3935 . . 3 {𝑦 ∈ Top ∣ 𝑥 = 𝑦} ⊆ 𝒫 𝒫 𝑥
92, 8ssexi 5215 . 2 {𝑦 ∈ Top ∣ 𝑥 = 𝑦} ∈ V
10 df-topon 21808 . 2 TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = 𝑦})
119, 10dmmpti 6522 1 dom TopOn = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1543  {cab 2714  {crab 3065  Vcvv 3408  𝒫 cpw 4513   cuni 4819  dom cdm 5551  Topctop 21790  TopOnctopon 21807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-fun 6382  df-fn 6383  df-topon 21808
This theorem is referenced by:  fntopon  21821
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