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Theorem dmi 4572
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 3268 . 2  |-  ( dom 
_I  =  _V  <->  A. x  x  e.  dom  _I  )
2 a9ev 1628 . . . 4  |-  E. y 
y  =  x
3 vex 2605 . . . . . . 7  |-  y  e. 
_V
43ideq 4510 . . . . . 6  |-  ( x  _I  y  <->  x  =  y )
5 equcom 1634 . . . . . 6  |-  ( x  =  y  <->  y  =  x )
64, 5bitri 182 . . . . 5  |-  ( x  _I  y  <->  y  =  x )
76exbii 1537 . . . 4  |-  ( E. y  x  _I  y  <->  E. y  y  =  x )
82, 7mpbir 144 . . 3  |-  E. y  x  _I  y
9 vex 2605 . . . 4  |-  x  e. 
_V
109eldm 4554 . . 3  |-  ( x  e.  dom  _I  <->  E. y  x  _I  y )
118, 10mpbir 144 . 2  |-  x  e. 
dom  _I
121, 11mpgbir 1383 1  |-  dom  _I  =  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1285   E.wex 1422    e. wcel 1434   _Vcvv 2602   class class class wbr 3787    _I cid 4045   dom cdm 4365
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 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-opab 3842  df-id 4050  df-xp 4371  df-rel 4372  df-dm 4375
This theorem is referenced by:  dmv  4573  iprc  4622  dmresi  4685  climshft2  10272
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