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Theorem dmsnsnsng 4986
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 2663 . . . . . . 7  |-  x  e. 
_V
21opid 3693 . . . . . 6  |-  <. x ,  x >.  =  { { x } }
3 sneq 3508 . . . . . . 7  |-  ( x  =  A  ->  { x }  =  { A } )
43sneqd 3510 . . . . . 6  |-  ( x  =  A  ->  { {
x } }  =  { { A } }
)
52, 4syl5eq 2162 . . . . 5  |-  ( x  =  A  ->  <. x ,  x >.  =  { { A } } )
65sneqd 3510 . . . 4  |-  ( x  =  A  ->  { <. x ,  x >. }  =  { { { A } } } )
76dmeqd 4711 . . 3  |-  ( x  =  A  ->  dom  {
<. x ,  x >. }  =  dom  { { { A } } }
)
87, 3eqeq12d 2132 . 2  |-  ( x  =  A  ->  ( dom  { <. x ,  x >. }  =  { x } 
<->  dom  { { { A } } }  =  { A } ) )
91dmsnop 4982 . 2  |-  dom  { <. x ,  x >. }  =  { x }
108, 9vtoclg 2720 1  |-  ( A  e.  _V  ->  dom  { { { A } } }  =  { A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    e. wcel 1465   _Vcvv 2660   {csn 3497   <.cop 3500   dom cdm 4509
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-dm 4519
This theorem is referenced by: (None)
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