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Theorem dmtopon 20775
Description: The domain of TopOn is 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 4879 . . . 4 𝒫 𝑥 ∈ V
21pwex 4878 . . 3 𝒫 𝒫 𝑥 ∈ V
3 eqcom 2658 . . . . 5 (𝑥 = 𝑦 𝑦 = 𝑥)
43rabbii 3216 . . . 4 {𝑦 ∈ Top ∣ 𝑥 = 𝑦} = {𝑦 ∈ Top ∣ 𝑦 = 𝑥}
5 rabssab 3723 . . . . 5 {𝑦 ∈ Top ∣ 𝑦 = 𝑥} ⊆ {𝑦 𝑦 = 𝑥}
6 pwpwssunieq 4647 . . . . 5 {𝑦 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥
75, 6sstri 3645 . . . 4 {𝑦 ∈ Top ∣ 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥
84, 7eqsstri 3668 . . 3 {𝑦 ∈ Top ∣ 𝑥 = 𝑦} ⊆ 𝒫 𝒫 𝑥
92, 8ssexi 4836 . 2 {𝑦 ∈ Top ∣ 𝑥 = 𝑦} ∈ V
10 df-topon 20764 . 2 TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = 𝑦})
119, 10dmmpti 6061 1 dom TopOn = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  {cab 2637  {crab 2945  Vcvv 3231  𝒫 cpw 4191   cuni 4468  dom cdm 5143  Topctop 20746  TopOnctopon 20763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-fun 5928  df-fn 5929  df-topon 20764
This theorem is referenced by:  fntopon  20776
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