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Theorem dmsnsnsn 5151
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 2791 . . . . . . . 8  |-  x  e. 
_V
21opid 3814 . . . . . . 7  |-  <. x ,  x >.  =  { { x } }
3 sneq 3651 . . . . . . . 8  |-  ( x  =  A  ->  { x }  =  { A } )
43sneqd 3653 . . . . . . 7  |-  ( x  =  A  ->  { {
x } }  =  { { A } }
)
52, 4syl5eq 2327 . . . . . 6  |-  ( x  =  A  ->  <. x ,  x >.  =  { { A } } )
65sneqd 3653 . . . . 5  |-  ( x  =  A  ->  { <. x ,  x >. }  =  { { { A } } } )
76dmeqd 4881 . . . 4  |-  ( x  =  A  ->  dom  {
<. x ,  x >. }  =  dom  { { { A } } }
)
87, 3eqeq12d 2297 . . 3  |-  ( x  =  A  ->  ( dom  { <. x ,  x >. }  =  { x } 
<->  dom  { { { A } } }  =  { A } ) )
91dmsnop 5147 . . 3  |-  dom  { <. x ,  x >. }  =  { x }
108, 9vtoclg 2843 . 2  |-  ( A  e.  _V  ->  dom  { { { A } } }  =  { A } )
11 0ex 4150 . . . . 5  |-  (/)  e.  _V
1211snid 3667 . . . 4  |-  (/)  e.  { (/)
}
13 dmsn0el 5142 . . . 4  |-  ( (/)  e.  { (/) }  ->  dom  { { (/) } }  =  (/) )
1412, 13ax-mp 8 . . 3  |-  dom  { { (/) } }  =  (/)
15 snprc 3695 . . . . . . 7  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
1615biimpi 186 . . . . . 6  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
1716sneqd 3653 . . . . 5  |-  ( -.  A  e.  _V  ->  { { A } }  =  { (/) } )
1817sneqd 3653 . . . 4  |-  ( -.  A  e.  _V  ->  { { { A } } }  =  { { (/) } } )
1918dmeqd 4881 . . 3  |-  ( -.  A  e.  _V  ->  dom 
{ { { A } } }  =  dom  { { (/) } } )
2014, 19, 163eqtr4a 2341 . 2  |-  ( -.  A  e.  _V  ->  dom 
{ { { A } } }  =  { A } )
2110, 20pm2.61i 156 1  |-  dom  { { { A } } }  =  { A }
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   {csn 3640   <.cop 3643   dom cdm 4689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-dm 4699
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