MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmsnsnsn Structured version   Unicode version

Theorem dmsnsnsn 5492
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.
StepHypRef Expression
1 vex 3112 . . . . . . . 8  |-  x  e. 
_V
21opid 4238 . . . . . . 7  |-  <. x ,  x >.  =  { { x } }
3 sneq 4042 . . . . . . . 8  |-  ( x  =  A  ->  { x }  =  { A } )
43sneqd 4044 . . . . . . 7  |-  ( x  =  A  ->  { {
x } }  =  { { A } }
)
52, 4syl5eq 2510 . . . . . 6  |-  ( x  =  A  ->  <. x ,  x >.  =  { { A } } )
65sneqd 4044 . . . . 5  |-  ( x  =  A  ->  { <. x ,  x >. }  =  { { { A } } } )
76dmeqd 5215 . . . 4  |-  ( x  =  A  ->  dom  {
<. x ,  x >. }  =  dom  { { { A } } }
)
87, 3eqeq12d 2479 . . 3  |-  ( x  =  A  ->  ( dom  { <. x ,  x >. }  =  { x } 
<->  dom  { { { A } } }  =  { A } ) )
91dmsnop 5488 . . 3  |-  dom  { <. x ,  x >. }  =  { x }
108, 9vtoclg 3167 . 2  |-  ( A  e.  _V  ->  dom  { { { A } } }  =  { A } )
11 0ex 4587 . . . . 5  |-  (/)  e.  _V
1211snid 4060 . . . 4  |-  (/)  e.  { (/)
}
13 dmsn0el 5483 . . . 4  |-  ( (/)  e.  { (/) }  ->  dom  { { (/) } }  =  (/) )
1412, 13ax-mp 5 . . 3  |-  dom  { { (/) } }  =  (/)
15 snprc 4095 . . . . . . 7  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
1615biimpi 194 . . . . . 6  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
1716sneqd 4044 . . . . 5  |-  ( -.  A  e.  _V  ->  { { A } }  =  { (/) } )
1817sneqd 4044 . . . 4  |-  ( -.  A  e.  _V  ->  { { { A } } }  =  { { (/) } } )
1918dmeqd 5215 . . 3  |-  ( -.  A  e.  _V  ->  dom 
{ { { A } } }  =  dom  { { (/) } } )
2014, 19, 163eqtr4a 2524 . 2  |-  ( -.  A  e.  _V  ->  dom 
{ { { A } } }  =  { A } )
2110, 20pm2.61i 164 1  |-  dom  { { { A } } }  =  { A }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1395    e. wcel 1819   _Vcvv 3109   (/)c0 3793   {csn 4032   <.cop 4038   dom cdm 5008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-dm 5018
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator