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Theorem dmi 4893
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 3470 . 2  |-  ( dom 
_I  =  _V  <->  A. x  x  e.  dom  _I  )
2 a9ev 1637 . . . 4  |-  E. y 
y  =  x
3 vex 2791 . . . . . . 7  |-  y  e. 
_V
43ideq 4836 . . . . . 6  |-  ( x  _I  y  <->  x  =  y )
5 equcom 1647 . . . . . 6  |-  ( x  =  y  <->  y  =  x )
64, 5bitri 240 . . . . 5  |-  ( x  _I  y  <->  y  =  x )
76exbii 1569 . . . 4  |-  ( E. y  x  _I  y  <->  E. y  y  =  x )
82, 7mpbir 200 . . 3  |-  E. y  x  _I  y
9 vex 2791 . . . 4  |-  x  e. 
_V
109eldm 4876 . . 3  |-  ( x  e.  dom  _I  <->  E. y  x  _I  y )
118, 10mpbir 200 . 2  |-  x  e. 
dom  _I
121, 11mpgbir 1537 1  |-  dom  _I  =  _V
Colors of variables: wff set class
Syntax hints:   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788   class class class wbr 4023    _I cid 4304   dom cdm 4689
This theorem is referenced by:  dmv  4894  iprc  4943  dmresi  5005
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-dm 4699
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