Theorem dmsnsnsn 5613

Description: The domain of the singleton of the singleton of a singleton. (Contributed by NM, 15-Sep-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
dmsnsnsn	|- dom { { { A } } } = { A }

Proof of Theorem dmsnsnsn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . . . . . 8 |- x e. _V
2	1	opid 4421	. . . . . . 7 |- <. x , x >. = { { x } }
3		sneq 4187	. . . . . . . 8 |- ( x = A -> { x } = { A } )
4	3	sneqd 4189	. . . . . . 7 |- ( x = A -> { { x } } = { { A } } )
5	2, 4	syl5eq 2668	. . . . . 6 |- ( x = A -> <. x , x >. = { { A } } )
6	5	sneqd 4189	. . . . 5 |- ( x = A -> { <. x , x >. } = { { { A } } } )
7	6	dmeqd 5326	. . . 4 |- ( x = A -> dom { <. x , x >. } = dom { { { A } } } )
8	7, 3	eqeq12d 2637	. . 3 |- ( x = A -> ( dom { <. x , x >. } = { x } <-> dom { { { A } } } = { A } ) )
9	1	dmsnop 5609	. . 3 |- dom { <. x , x >. } = { x }
10	8, 9	vtoclg 3266	. 2 |- ( A e. _V -> dom { { { A } } } = { A } )
11		0ex 4790	. . . . 5 |- (/) e. _V
12	11	snid 4208	. . . 4 |- (/) e. { (/) }
13		dmsn0el 5604	. . . 4 |- ( (/) e. { (/) } -> dom { { (/) } } = (/) )
14	12, 13	ax-mp 5	. . 3 |- dom { { (/) } } = (/)
15		snprc 4253	. . . . . . 7 |- ( -. A e. _V <-> { A } = (/) )
16	15	biimpi 206	. . . . . 6 |- ( -. A e. _V -> { A } = (/) )
17	16	sneqd 4189	. . . . 5 |- ( -. A e. _V -> { { A } } = { (/) } )
18	17	sneqd 4189	. . . 4 |- ( -. A e. _V -> { { { A } } } = { { (/) } } )
19	18	dmeqd 5326	. . 3 |- ( -. A e. _V -> dom { { { A } } } = dom { { (/) } } )
20	14, 19, 16	3eqtr4a 2682	. 2 |- ( -. A e. _V -> dom { { { A } } } = { A } )
21	10, 20	pm2.61i 176	1 |- dom { { { A } } } = { A }

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   {csn 4177   <.cop 4183   dom cdm 5114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-dm 5124

This theorem is referenced by: (None)