Theorem dmsnsnsng 4818

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.

Step	Hyp	Ref	Expression
1		vex 2604	. . . . . . 7 ⊢ 𝑥 ∈ V
2	1	opid 3588	. . . . . 6 ⊢ ⟨𝑥, 𝑥⟩ = {{𝑥}}
3		sneq 3409	. . . . . . 7 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
4	3	sneqd 3411	. . . . . 6 ⊢ (𝑥 = 𝐴 → {{𝑥}} = {{𝐴}})
5	2, 4	syl5eq 2125	. . . . 5 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑥⟩ = {{𝐴}})
6	5	sneqd 3411	. . . 4 ⊢ (𝑥 = 𝐴 → {⟨𝑥, 𝑥⟩} = {{{𝐴}}})
7	6	dmeqd 4555	. . 3 ⊢ (𝑥 = 𝐴 → dom {⟨𝑥, 𝑥⟩} = dom {{{𝐴}}})
8	7, 3	eqeq12d 2095	. 2 ⊢ (𝑥 = 𝐴 → (dom {⟨𝑥, 𝑥⟩} = {𝑥} ↔ dom {{{𝐴}}} = {𝐴}))
9	1	dmsnop 4814	. 2 ⊢ dom {⟨𝑥, 𝑥⟩} = {𝑥}
10	8, 9	vtoclg 2658	1 ⊢ (𝐴 ∈ V → dom {{{𝐴}}} = {𝐴})

Syntax hints:   → wi 4   = wceq 1284   ∈ wcel 1433  Vcvv 2601  {csn 3398  ⟨cop 3401  dom cdm 4363

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-dm 4373

This theorem is referenced by: (None)