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Theorem dmsnsnsng 4828
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 2605 . . . . . . 7  |-  x  e. 
_V
21opid 3596 . . . . . 6  |-  <. x ,  x >.  =  { { x } }
3 sneq 3417 . . . . . . 7  |-  ( x  =  A  ->  { x }  =  { A } )
43sneqd 3419 . . . . . 6  |-  ( x  =  A  ->  { {
x } }  =  { { A } }
)
52, 4syl5eq 2126 . . . . 5  |-  ( x  =  A  ->  <. x ,  x >.  =  { { A } } )
65sneqd 3419 . . . 4  |-  ( x  =  A  ->  { <. x ,  x >. }  =  { { { A } } } )
76dmeqd 4565 . . 3  |-  ( x  =  A  ->  dom  {
<. x ,  x >. }  =  dom  { { { A } } }
)
87, 3eqeq12d 2096 . 2  |-  ( x  =  A  ->  ( dom  { <. x ,  x >. }  =  { x } 
<->  dom  { { { A } } }  =  { A } ) )
91dmsnop 4824 . 2  |-  dom  { <. x ,  x >. }  =  { x }
108, 9vtoclg 2659 1  |-  ( A  e.  _V  ->  dom  { { { A } } }  =  { A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434   _Vcvv 2602   {csn 3406   <.cop 3409   dom cdm 4371
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-dm 4381
This theorem is referenced by: (None)
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