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Theorem dmtopon 21535
 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 5243 . . . 4 𝒫 𝑥 ∈ V
21pwex 5246 . . 3 𝒫 𝒫 𝑥 ∈ V
3 eqcom 2805 . . . . 5 (𝑥 = 𝑦 𝑦 = 𝑥)
43rabbii 3420 . . . 4 {𝑦 ∈ Top ∣ 𝑥 = 𝑦} = {𝑦 ∈ Top ∣ 𝑦 = 𝑥}
5 rabssab 4011 . . . . 5 {𝑦 ∈ Top ∣ 𝑦 = 𝑥} ⊆ {𝑦 𝑦 = 𝑥}
6 pwpwssunieq 4989 . . . . 5 {𝑦 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥
75, 6sstri 3924 . . . 4 {𝑦 ∈ Top ∣ 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥
84, 7eqsstri 3949 . . 3 {𝑦 ∈ Top ∣ 𝑥 = 𝑦} ⊆ 𝒫 𝒫 𝑥
92, 8ssexi 5190 . 2 {𝑦 ∈ Top ∣ 𝑥 = 𝑦} ∈ V
10 df-topon 21523 . 2 TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = 𝑦})
119, 10dmmpti 6464 1 dom TopOn = V
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538  {cab 2776  {crab 3110  Vcvv 3441  𝒫 cpw 4497  ∪ cuni 4800  dom cdm 5519  Topctop 21505  TopOnctopon 21522 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-fun 6326  df-fn 6327  df-topon 21523 This theorem is referenced by:  fntopon  21536
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