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Theorem dmsnsnsng 4895
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 2622 . . . . . . 7  |-  x  e. 
_V
21opid 3635 . . . . . 6  |-  <. x ,  x >.  =  { { x } }
3 sneq 3452 . . . . . . 7  |-  ( x  =  A  ->  { x }  =  { A } )
43sneqd 3454 . . . . . 6  |-  ( x  =  A  ->  { {
x } }  =  { { A } }
)
52, 4syl5eq 2132 . . . . 5  |-  ( x  =  A  ->  <. x ,  x >.  =  { { A } } )
65sneqd 3454 . . . 4  |-  ( x  =  A  ->  { <. x ,  x >. }  =  { { { A } } } )
76dmeqd 4626 . . 3  |-  ( x  =  A  ->  dom  {
<. x ,  x >. }  =  dom  { { { A } } }
)
87, 3eqeq12d 2102 . 2  |-  ( x  =  A  ->  ( dom  { <. x ,  x >. }  =  { x } 
<->  dom  { { { A } } }  =  { A } ) )
91dmsnop 4891 . 2  |-  dom  { <. x ,  x >. }  =  { x }
108, 9vtoclg 2679 1  |-  ( A  e.  _V  ->  dom  { { { A } } }  =  { A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438   _Vcvv 2619   {csn 3441   <.cop 3444   dom cdm 4428
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-dm 4438
This theorem is referenced by: (None)
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