Theorem dmsnsnsng 5081

Description: The domain of the singleton of the singleton of a singleton. (Contributed by Jim Kingdon, 16-Dec-2018.)

Assertion

Ref	Expression
dmsnsnsng	|- ( A e. _V -> dom { { { A } } } = { A } )

Proof of Theorem dmsnsnsng

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2729	. . . . . . 7 |- x e. _V
2	1	opid 3776	. . . . . 6 |- <. x , x >. = { { x } }
3		sneq 3587	. . . . . . 7 |- ( x = A -> { x } = { A } )
4	3	sneqd 3589	. . . . . 6 |- ( x = A -> { { x } } = { { A } } )
5	2, 4	syl5eq 2211	. . . . 5 |- ( x = A -> <. x , x >. = { { A } } )
6	5	sneqd 3589	. . . 4 |- ( x = A -> { <. x , x >. } = { { { A } } } )
7	6	dmeqd 4806	. . 3 |- ( x = A -> dom { <. x , x >. } = dom { { { A } } } )
8	7, 3	eqeq12d 2180	. 2 |- ( x = A -> ( dom { <. x , x >. } = { x } <-> dom { { { A } } } = { A } ) )
9	1	dmsnop 5077	. 2 |- dom { <. x , x >. } = { x }
10	8, 9	vtoclg 2786	1 |- ( A e. _V -> dom { { { A } } } = { A } )

Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136   _Vcvv 2726   {csn 3576   <.cop 3579   dom cdm 4604

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-dm 4614

This theorem is referenced by: (None)