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Theorem bj-dmtopon 32242
 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 4775 . . . 4 𝒫 𝑥 ∈ V
21pwex 4774 . . 3 𝒫 𝒫 𝑥 ∈ V
3 eqcom 2617 . . . . . 6 (𝑥 = 𝑦 𝑦 = 𝑥)
43a1i 11 . . . . 5 (𝑦 ∈ Top → (𝑥 = 𝑦 𝑦 = 𝑥))
54rabbiia 3161 . . . 4 {𝑦 ∈ Top ∣ 𝑥 = 𝑦} = {𝑦 ∈ Top ∣ 𝑦 = 𝑥}
6 rabssab 3652 . . . . 5 {𝑦 ∈ Top ∣ 𝑦 = 𝑥} ⊆ {𝑦 𝑦 = 𝑥}
7 bj-sspwpweq 32240 . . . . 5 {𝑦 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥
86, 7sstri 3577 . . . 4 {𝑦 ∈ Top ∣ 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥
95, 8eqsstri 3598 . . 3 {𝑦 ∈ Top ∣ 𝑥 = 𝑦} ⊆ 𝒫 𝒫 𝑥
102, 9ssexi 4731 . 2 {𝑦 ∈ Top ∣ 𝑥 = 𝑦} ∈ V
11 df-topon 20523 . 2 TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = 𝑦})
1210, 11dmmpti 5936 1 dom TopOn = V
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {cab 2596  {crab 2900  Vcvv 3173  𝒫 cpw 4108  ∪ cuni 4372  dom cdm 5038  Topctop 20517  TopOnctopon 20518 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-fun 5806  df-fn 5807  df-topon 20523 This theorem is referenced by:  bj-fntopon  32243
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