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Theorem dmsnsnsn 5486
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 3116 . . . . . . . 8  |-  x  e. 
_V
21opid 4232 . . . . . . 7  |-  <. x ,  x >.  =  { { x } }
3 sneq 4037 . . . . . . . 8  |-  ( x  =  A  ->  { x }  =  { A } )
43sneqd 4039 . . . . . . 7  |-  ( x  =  A  ->  { {
x } }  =  { { A } }
)
52, 4syl5eq 2520 . . . . . 6  |-  ( x  =  A  ->  <. x ,  x >.  =  { { A } } )
65sneqd 4039 . . . . 5  |-  ( x  =  A  ->  { <. x ,  x >. }  =  { { { A } } } )
76dmeqd 5205 . . . 4  |-  ( x  =  A  ->  dom  {
<. x ,  x >. }  =  dom  { { { A } } }
)
87, 3eqeq12d 2489 . . 3  |-  ( x  =  A  ->  ( dom  { <. x ,  x >. }  =  { x } 
<->  dom  { { { A } } }  =  { A } ) )
91dmsnop 5482 . . 3  |-  dom  { <. x ,  x >. }  =  { x }
108, 9vtoclg 3171 . 2  |-  ( A  e.  _V  ->  dom  { { { A } } }  =  { A } )
11 0ex 4577 . . . . 5  |-  (/)  e.  _V
1211snid 4055 . . . 4  |-  (/)  e.  { (/)
}
13 dmsn0el 5477 . . . 4  |-  ( (/)  e.  { (/) }  ->  dom  { { (/) } }  =  (/) )
1412, 13ax-mp 5 . . 3  |-  dom  { { (/) } }  =  (/)
15 snprc 4091 . . . . . . 7  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
1615biimpi 194 . . . . . 6  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
1716sneqd 4039 . . . . 5  |-  ( -.  A  e.  _V  ->  { { A } }  =  { (/) } )
1817sneqd 4039 . . . 4  |-  ( -.  A  e.  _V  ->  { { { A } } }  =  { { (/) } } )
1918dmeqd 5205 . . 3  |-  ( -.  A  e.  _V  ->  dom 
{ { { A } } }  =  dom  { { (/) } } )
2014, 19, 163eqtr4a 2534 . 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 1379    e. wcel 1767   _Vcvv 3113   (/)c0 3785   {csn 4027   <.cop 4033   dom cdm 4999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-dm 5009
This theorem is referenced by: (None)
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