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Theorem dmsnsnsn 5281
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 2895 . . . . . . . 8  |-  x  e. 
_V
21opid 3937 . . . . . . 7  |-  <. x ,  x >.  =  { { x } }
3 sneq 3761 . . . . . . . 8  |-  ( x  =  A  ->  { x }  =  { A } )
43sneqd 3763 . . . . . . 7  |-  ( x  =  A  ->  { {
x } }  =  { { A } }
)
52, 4syl5eq 2424 . . . . . 6  |-  ( x  =  A  ->  <. x ,  x >.  =  { { A } } )
65sneqd 3763 . . . . 5  |-  ( x  =  A  ->  { <. x ,  x >. }  =  { { { A } } } )
76dmeqd 5005 . . . 4  |-  ( x  =  A  ->  dom  {
<. x ,  x >. }  =  dom  { { { A } } }
)
87, 3eqeq12d 2394 . . 3  |-  ( x  =  A  ->  ( dom  { <. x ,  x >. }  =  { x } 
<->  dom  { { { A } } }  =  { A } ) )
91dmsnop 5277 . . 3  |-  dom  { <. x ,  x >. }  =  { x }
108, 9vtoclg 2947 . 2  |-  ( A  e.  _V  ->  dom  { { { A } } }  =  { A } )
11 0ex 4273 . . . . 5  |-  (/)  e.  _V
1211snid 3777 . . . 4  |-  (/)  e.  { (/)
}
13 dmsn0el 5272 . . . 4  |-  ( (/)  e.  { (/) }  ->  dom  { { (/) } }  =  (/) )
1412, 13ax-mp 8 . . 3  |-  dom  { { (/) } }  =  (/)
15 snprc 3807 . . . . . . 7  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
1615biimpi 187 . . . . . 6  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
1716sneqd 3763 . . . . 5  |-  ( -.  A  e.  _V  ->  { { A } }  =  { (/) } )
1817sneqd 3763 . . . 4  |-  ( -.  A  e.  _V  ->  { { { A } } }  =  { { (/) } } )
1918dmeqd 5005 . . 3  |-  ( -.  A  e.  _V  ->  dom 
{ { { A } } }  =  dom  { { (/) } } )
2014, 19, 163eqtr4a 2438 . 2  |-  ( -.  A  e.  _V  ->  dom 
{ { { A } } }  =  { A } )
2110, 20pm2.61i 158 1  |-  dom  { { { A } } }  =  { A }
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1649    e. wcel 1717   _Vcvv 2892   (/)c0 3564   {csn 3750   <.cop 3753   dom cdm 4811
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-xp 4817  df-dm 4821
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