Theorem dmsnsnsng 5240

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 2816	. . . . . . 7 ⊢ 𝑥 ∈ V
2	1	opid 3901	. . . . . 6 ⊢ ⟨𝑥, 𝑥⟩ = {{𝑥}}
3		sneq 3700	. . . . . . 7 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
4	3	sneqd 3702	. . . . . 6 ⊢ (𝑥 = 𝐴 → {{𝑥}} = {{𝐴}})
5	2, 4	eqtrid 2277	. . . . 5 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑥⟩ = {{𝐴}})
6	5	sneqd 3702	. . . 4 ⊢ (𝑥 = 𝐴 → {⟨𝑥, 𝑥⟩} = {{{𝐴}}})
7	6	dmeqd 4958	. . 3 ⊢ (𝑥 = 𝐴 → dom {⟨𝑥, 𝑥⟩} = dom {{{𝐴}}})
8	7, 3	eqeq12d 2247	. 2 ⊢ (𝑥 = 𝐴 → (dom {⟨𝑥, 𝑥⟩} = {𝑥} ↔ dom {{{𝐴}}} = {𝐴}))
9	1	dmsnop 5236	. 2 ⊢ dom {⟨𝑥, 𝑥⟩} = {𝑥}
10	8, 9	vtoclg 2875	1 ⊢ (𝐴 ∈ V → dom {{{𝐴}}} = {𝐴})

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2203  Vcvv 2813  {csn 3689  ⟨cop 3692  dom cdm 4749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-dm 4759

This theorem is referenced by: (None)