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Theorem dmsnsnsn 5340
Description: The domain of the singleton of the singleton of a singleton. (Contributed by NM, 15-Sep-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
dmsnsnsn  |-  dom  { { { A } } }  =  { A }

Proof of Theorem dmsnsnsn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 vex 2951 . . . . . . . 8  |-  x  e. 
_V
21opid 3994 . . . . . . 7  |-  <. x ,  x >.  =  { { x } }
3 sneq 3817 . . . . . . . 8  |-  ( x  =  A  ->  { x }  =  { A } )
43sneqd 3819 . . . . . . 7  |-  ( x  =  A  ->  { {
x } }  =  { { A } }
)
52, 4syl5eq 2479 . . . . . 6  |-  ( x  =  A  ->  <. x ,  x >.  =  { { A } } )
65sneqd 3819 . . . . 5  |-  ( x  =  A  ->  { <. x ,  x >. }  =  { { { A } } } )
76dmeqd 5064 . . . 4  |-  ( x  =  A  ->  dom  {
<. x ,  x >. }  =  dom  { { { A } } }
)
87, 3eqeq12d 2449 . . 3  |-  ( x  =  A  ->  ( dom  { <. x ,  x >. }  =  { x } 
<->  dom  { { { A } } }  =  { A } ) )
91dmsnop 5336 . . 3  |-  dom  { <. x ,  x >. }  =  { x }
108, 9vtoclg 3003 . 2  |-  ( A  e.  _V  ->  dom  { { { A } } }  =  { A } )
11 0ex 4331 . . . . 5  |-  (/)  e.  _V
1211snid 3833 . . . 4  |-  (/)  e.  { (/)
}
13 dmsn0el 5331 . . . 4  |-  ( (/)  e.  { (/) }  ->  dom  { { (/) } }  =  (/) )
1412, 13ax-mp 8 . . 3  |-  dom  { { (/) } }  =  (/)
15 snprc 3863 . . . . . . 7  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
1615biimpi 187 . . . . . 6  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
1716sneqd 3819 . . . . 5  |-  ( -.  A  e.  _V  ->  { { A } }  =  { (/) } )
1817sneqd 3819 . . . 4  |-  ( -.  A  e.  _V  ->  { { { A } } }  =  { { (/) } } )
1918dmeqd 5064 . . 3  |-  ( -.  A  e.  _V  ->  dom 
{ { { A } } }  =  dom  { { (/) } } )
2014, 19, 163eqtr4a 2493 . 2  |-  ( -.  A  e.  _V  ->  dom 
{ { { A } } }  =  { A } )
2110, 20pm2.61i 158 1  |-  dom  { { { A } } }  =  { A }
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1652    e. wcel 1725   _Vcvv 2948   (/)c0 3620   {csn 3806   <.cop 3809   dom cdm 4870
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-dm 4880
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