Theorem dmsnsnsng 4798

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 2560	. . . . . . 7 |- x e. _V
2	1	opid 3567	. . . . . 6 |- <. x , x >. = { { x } }
3		sneq 3386	. . . . . . 7 |- ( x = A -> { x } = { A } )
4	3	sneqd 3388	. . . . . 6 |- ( x = A -> { { x } } = { { A } } )
5	2, 4	syl5eq 2084	. . . . 5 |- ( x = A -> <. x , x >. = { { A } } )
6	5	sneqd 3388	. . . 4 |- ( x = A -> { <. x , x >. } = { { { A } } } )
7	6	dmeqd 4537	. . 3 |- ( x = A -> dom { <. x , x >. } = dom { { { A } } } )
8	7, 3	eqeq12d 2054	. 2 |- ( x = A -> ( dom { <. x , x >. } = { x } <-> dom { { { A } } } = { A } ) )
9	1	dmsnop 4794	. 2 |- dom { <. x , x >. } = { x }
10	8, 9	vtoclg 2613	1 |- ( A e. _V -> dom { { { A } } } = { A } )

Syntax hints:   -> wi 4   = wceq 1243   e. wcel 1393   _Vcvv 2557   {csn 3375   <.cop 3378   dom cdm 4345

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-dm 4355

This theorem is referenced by: (None)