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Theorem bj-dmtopon 32690
Description: The domain of TopOn is V. (Contributed by BJ, 29-Apr-2021.)
Assertion
Ref Expression
bj-dmtopon dom TopOn = V

Proof of Theorem bj-dmtopon
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vpwex 4814 . . . 4 𝒫 𝑥 ∈ V
21pwex 4813 . . 3 𝒫 𝒫 𝑥 ∈ V
3 eqcom 2633 . . . . . 6 (𝑥 = 𝑦 𝑦 = 𝑥)
43a1i 11 . . . . 5 (𝑦 ∈ Top → (𝑥 = 𝑦 𝑦 = 𝑥))
54rabbiia 3178 . . . 4 {𝑦 ∈ Top ∣ 𝑥 = 𝑦} = {𝑦 ∈ Top ∣ 𝑦 = 𝑥}
6 rabssab 3673 . . . . 5 {𝑦 ∈ Top ∣ 𝑦 = 𝑥} ⊆ {𝑦 𝑦 = 𝑥}
7 bj-sspwpweq 32688 . . . . 5 {𝑦 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥
86, 7sstri 3597 . . . 4 {𝑦 ∈ Top ∣ 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥
95, 8eqsstri 3619 . . 3 {𝑦 ∈ Top ∣ 𝑥 = 𝑦} ⊆ 𝒫 𝒫 𝑥
102, 9ssexi 4768 . 2 {𝑦 ∈ Top ∣ 𝑥 = 𝑦} ∈ V
11 df-topon 20618 . 2 TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = 𝑦})
1210, 11dmmpti 5982 1 dom TopOn = V
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  wcel 1992  {cab 2612  {crab 2916  Vcvv 3191  𝒫 cpw 4135   cuni 4407  dom cdm 5079  Topctop 20612  TopOnctopon 20613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-fun 5852  df-fn 5853  df-topon 20618
This theorem is referenced by:  bj-fntopon  32691
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