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Theorem dmi 4664
Description: The domain of the identity relation is the universe. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dmi  |-  dom  _I  =  _V

Proof of Theorem dmi
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqv 3306 . 2  |-  ( dom 
_I  =  _V  <->  A. x  x  e.  dom  _I  )
2 a9ev 1633 . . . 4  |-  E. y 
y  =  x
3 vex 2623 . . . . . . 7  |-  y  e. 
_V
43ideq 4601 . . . . . 6  |-  ( x  _I  y  <->  x  =  y )
5 equcom 1640 . . . . . 6  |-  ( x  =  y  <->  y  =  x )
64, 5bitri 183 . . . . 5  |-  ( x  _I  y  <->  y  =  x )
76exbii 1542 . . . 4  |-  ( E. y  x  _I  y  <->  E. y  y  =  x )
82, 7mpbir 145 . . 3  |-  E. y  x  _I  y
9 vex 2623 . . . 4  |-  x  e. 
_V
109eldm 4646 . . 3  |-  ( x  e.  dom  _I  <->  E. y  x  _I  y )
118, 10mpbir 145 . 2  |-  x  e. 
dom  _I
121, 11mpgbir 1388 1  |-  dom  _I  =  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1290   E.wex 1427    e. wcel 1439   _Vcvv 2620   class class class wbr 3851    _I cid 4124   dom cdm 4452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-id 4129  df-xp 4458  df-rel 4459  df-dm 4462
This theorem is referenced by:  dmv  4665  iprc  4714  dmresi  4780  climshft2  10756
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