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Theorem dmi 5048
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 3747 . 2  |-  ( dom 
_I  =  _V  <->  A. x  x  e.  dom  _I  )
2 ax6ev 1806 . . . 4  |-  E. y 
y  =  x
3 vex 3047 . . . . . . 7  |-  y  e. 
_V
43ideq 4986 . . . . . 6  |-  ( x  _I  y  <->  x  =  y )
5 equcom 1861 . . . . . 6  |-  ( x  =  y  <->  y  =  x )
64, 5bitri 253 . . . . 5  |-  ( x  _I  y  <->  y  =  x )
76exbii 1717 . . . 4  |-  ( E. y  x  _I  y  <->  E. y  y  =  x )
82, 7mpbir 213 . . 3  |-  E. y  x  _I  y
9 vex 3047 . . . 4  |-  x  e. 
_V
109eldm 5031 . . 3  |-  ( x  e.  dom  _I  <->  E. y  x  _I  y )
118, 10mpbir 213 . 2  |-  x  e. 
dom  _I
121, 11mpgbir 1672 1  |-  dom  _I  =  _V
Colors of variables: wff setvar class
Syntax hints:    = wceq 1443   E.wex 1662    e. wcel 1886   _Vcvv 3044   class class class wbr 4401    _I cid 4743   dom cdm 4833
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pr 4638
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-br 4402  df-opab 4461  df-id 4748  df-xp 4839  df-rel 4840  df-dm 4843
This theorem is referenced by:  dmv  5049  dmresi  5159  iprc  6725
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