Theorem dmsnsnsng 5148

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 2766	. . . . . . 7 |- x e. _V
2	1	opid 3827	. . . . . 6 |- <. x , x >. = { { x } }
3		sneq 3634	. . . . . . 7 |- ( x = A -> { x } = { A } )
4	3	sneqd 3636	. . . . . 6 |- ( x = A -> { { x } } = { { A } } )
5	2, 4	eqtrid 2241	. . . . 5 |- ( x = A -> <. x , x >. = { { A } } )
6	5	sneqd 3636	. . . 4 |- ( x = A -> { <. x , x >. } = { { { A } } } )
7	6	dmeqd 4869	. . 3 |- ( x = A -> dom { <. x , x >. } = dom { { { A } } } )
8	7, 3	eqeq12d 2211	. 2 |- ( x = A -> ( dom { <. x , x >. } = { x } <-> dom { { { A } } } = { A } ) )
9	1	dmsnop 5144	. 2 |- dom { <. x , x >. } = { x }
10	8, 9	vtoclg 2824	1 |- ( A e. _V -> dom { { { A } } } = { A } )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   _Vcvv 2763   {csn 3623   <.cop 3626   dom cdm 4664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-dm 4674

This theorem is referenced by: (None)