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Theorem dmtopon 12227
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 4110 . . . 4 𝒫 𝑥 ∈ V
21pwex 4114 . . 3 𝒫 𝒫 𝑥 ∈ V
3 eqcom 2142 . . . . 5 (𝑥 = 𝑦 𝑦 = 𝑥)
43rabbii 2675 . . . 4 {𝑦 ∈ Top ∣ 𝑥 = 𝑦} = {𝑦 ∈ Top ∣ 𝑦 = 𝑥}
5 rabssab 3188 . . . . 5 {𝑦 ∈ Top ∣ 𝑦 = 𝑥} ⊆ {𝑦 𝑦 = 𝑥}
6 pwpwssunieq 3908 . . . . 5 {𝑦 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥
75, 6sstri 3110 . . . 4 {𝑦 ∈ Top ∣ 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥
84, 7eqsstri 3133 . . 3 {𝑦 ∈ Top ∣ 𝑥 = 𝑦} ⊆ 𝒫 𝒫 𝑥
92, 8ssexi 4073 . 2 {𝑦 ∈ Top ∣ 𝑥 = 𝑦} ∈ V
10 df-topon 12215 . 2 TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = 𝑦})
119, 10dmmpti 5259 1 dom TopOn = V
Colors of variables: wff set class
Syntax hints:   = wceq 1332  {cab 2126  {crab 2421  Vcvv 2689  𝒫 cpw 3514   cuni 3743  dom cdm 4546  Topctop 12201  TopOnctopon 12214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-fun 5132  df-fn 5133  df-topon 12215
This theorem is referenced by:  fntopon  12228
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