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Theorem dmsnsnsng 5076
Description: The domain of the singleton of the singleton of a singleton. (Contributed by Jim Kingdon, 16-Dec-2018.)
Assertion
Ref Expression
dmsnsnsng  |-  ( A  e.  _V  ->  dom  { { { A } } }  =  { A } )

Proof of Theorem dmsnsnsng
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 vex 2725 . . . . . . 7  |-  x  e. 
_V
21opid 3771 . . . . . 6  |-  <. x ,  x >.  =  { { x } }
3 sneq 3582 . . . . . . 7  |-  ( x  =  A  ->  { x }  =  { A } )
43sneqd 3584 . . . . . 6  |-  ( x  =  A  ->  { {
x } }  =  { { A } }
)
52, 4syl5eq 2209 . . . . 5  |-  ( x  =  A  ->  <. x ,  x >.  =  { { A } } )
65sneqd 3584 . . . 4  |-  ( x  =  A  ->  { <. x ,  x >. }  =  { { { A } } } )
76dmeqd 4801 . . 3  |-  ( x  =  A  ->  dom  {
<. x ,  x >. }  =  dom  { { { A } } }
)
87, 3eqeq12d 2179 . 2  |-  ( x  =  A  ->  ( dom  { <. x ,  x >. }  =  { x } 
<->  dom  { { { A } } }  =  { A } ) )
91dmsnop 5072 . 2  |-  dom  { <. x ,  x >. }  =  { x }
108, 9vtoclg 2782 1  |-  ( A  e.  _V  ->  dom  { { { A } } }  =  { A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1342    e. wcel 2135   _Vcvv 2722   {csn 3571   <.cop 3574   dom cdm 4599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-pow 4148  ax-pr 4182
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2724  df-un 3116  df-in 3118  df-ss 3125  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-br 3978  df-dm 4609
This theorem is referenced by: (None)
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