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 e. _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 e. _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 e. _V -> dom { { { A } } } = { A } )

Syntax hints:   -> wi 4   = wceq 1284   e. 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)