ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dmsnsnsng Unicode version

Theorem dmsnsnsng 5121
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 2755 . . . . . . 7  |-  x  e. 
_V
21opid 3811 . . . . . 6  |-  <. x ,  x >.  =  { { x } }
3 sneq 3618 . . . . . . 7  |-  ( x  =  A  ->  { x }  =  { A } )
43sneqd 3620 . . . . . 6  |-  ( x  =  A  ->  { {
x } }  =  { { A } }
)
52, 4eqtrid 2234 . . . . 5  |-  ( x  =  A  ->  <. x ,  x >.  =  { { A } } )
65sneqd 3620 . . . 4  |-  ( x  =  A  ->  { <. x ,  x >. }  =  { { { A } } } )
76dmeqd 4844 . . 3  |-  ( x  =  A  ->  dom  {
<. x ,  x >. }  =  dom  { { { A } } }
)
87, 3eqeq12d 2204 . 2  |-  ( x  =  A  ->  ( dom  { <. x ,  x >. }  =  { x } 
<->  dom  { { { A } } }  =  { A } ) )
91dmsnop 5117 . 2  |-  dom  { <. x ,  x >. }  =  { x }
108, 9vtoclg 2812 1  |-  ( A  e.  _V  ->  dom  { { { A } } }  =  { A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 2160   _Vcvv 2752   {csn 3607   <.cop 3610   dom cdm 4641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-dm 4651
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator