Theorem dmsnsnsng 4895

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 2622	. . . . . . 7 |- x e. _V
2	1	opid 3635	. . . . . 6 |- <. x , x >. = { { x } }
3		sneq 3452	. . . . . . 7 |- ( x = A -> { x } = { A } )
4	3	sneqd 3454	. . . . . 6 |- ( x = A -> { { x } } = { { A } } )
5	2, 4	syl5eq 2132	. . . . 5 |- ( x = A -> <. x , x >. = { { A } } )
6	5	sneqd 3454	. . . 4 |- ( x = A -> { <. x , x >. } = { { { A } } } )
7	6	dmeqd 4626	. . 3 |- ( x = A -> dom { <. x , x >. } = dom { { { A } } } )
8	7, 3	eqeq12d 2102	. 2 |- ( x = A -> ( dom { <. x , x >. } = { x } <-> dom { { { A } } } = { A } ) )
9	1	dmsnop 4891	. 2 |- dom { <. x , x >. } = { x }
10	8, 9	vtoclg 2679	1 |- ( A e. _V -> dom { { { A } } } = { A } )

Syntax hints:   -> wi 4   = wceq 1289   e. wcel 1438   _Vcvv 2619   {csn 3441   <.cop 3444   dom cdm 4428

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 3949  ax-pow 4001  ax-pr 4027

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 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-dm 4438

This theorem is referenced by: (None)