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Theorem dmsnsnsng 4908
Description: The domain of the singleton of the singleton of a singleton. (Contributed by Jim Kingdon, 16-Dec-2018.)
Assertion
Ref Expression
dmsnsnsng (𝐴 ∈ V → dom {{{𝐴}}} = {𝐴})

Proof of Theorem dmsnsnsng
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 vex 2622 . . . . . . 7 𝑥 ∈ V
21opid 3640 . . . . . 6 𝑥, 𝑥⟩ = {{𝑥}}
3 sneq 3457 . . . . . . 7 (𝑥 = 𝐴 → {𝑥} = {𝐴})
43sneqd 3459 . . . . . 6 (𝑥 = 𝐴 → {{𝑥}} = {{𝐴}})
52, 4syl5eq 2132 . . . . 5 (𝑥 = 𝐴 → ⟨𝑥, 𝑥⟩ = {{𝐴}})
65sneqd 3459 . . . 4 (𝑥 = 𝐴 → {⟨𝑥, 𝑥⟩} = {{{𝐴}}})
76dmeqd 4638 . . 3 (𝑥 = 𝐴 → dom {⟨𝑥, 𝑥⟩} = dom {{{𝐴}}})
87, 3eqeq12d 2102 . 2 (𝑥 = 𝐴 → (dom {⟨𝑥, 𝑥⟩} = {𝑥} ↔ dom {{{𝐴}}} = {𝐴}))
91dmsnop 4904 . 2 dom {⟨𝑥, 𝑥⟩} = {𝑥}
108, 9vtoclg 2679 1 (𝐴 ∈ V → dom {{{𝐴}}} = {𝐴})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  wcel 1438  Vcvv 2619  {csn 3446  cop 3449  dom cdm 4438
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-dm 4448
This theorem is referenced by: (None)
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