Theorem dmsnsnsng 5024

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 2692	. . . . . . 7 ⊢ 𝑥 ∈ V
2	1	opid 3731	. . . . . 6 ⊢ ⟨𝑥, 𝑥⟩ = {{𝑥}}
3		sneq 3543	. . . . . . 7 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
4	3	sneqd 3545	. . . . . 6 ⊢ (𝑥 = 𝐴 → {{𝑥}} = {{𝐴}})
5	2, 4	syl5eq 2185	. . . . 5 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑥⟩ = {{𝐴}})
6	5	sneqd 3545	. . . 4 ⊢ (𝑥 = 𝐴 → {⟨𝑥, 𝑥⟩} = {{{𝐴}}})
7	6	dmeqd 4749	. . 3 ⊢ (𝑥 = 𝐴 → dom {⟨𝑥, 𝑥⟩} = dom {{{𝐴}}})
8	7, 3	eqeq12d 2155	. 2 ⊢ (𝑥 = 𝐴 → (dom {⟨𝑥, 𝑥⟩} = {𝑥} ↔ dom {{{𝐴}}} = {𝐴}))
9	1	dmsnop 5020	. 2 ⊢ dom {⟨𝑥, 𝑥⟩} = {𝑥}
10	8, 9	vtoclg 2749	1 ⊢ (𝐴 ∈ V → dom {{{𝐴}}} = {𝐴})

Syntax hints:   → wi 4   = wceq 1332   ∈ wcel 1481  Vcvv 2689  {csn 3532  ⟨cop 3535  dom cdm 4547

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 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-dm 4557

This theorem is referenced by: (None)