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Theorem dmi 4909
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 3483 . 2  |-  ( dom 
_I  =  _V  <->  A. x  x  e.  dom  _I  )
2 a9ev 1646 . . . 4  |-  E. y 
y  =  x
3 vex 2804 . . . . . . 7  |-  y  e. 
_V
43ideq 4852 . . . . . 6  |-  ( x  _I  y  <->  x  =  y )
5 equcom 1665 . . . . . 6  |-  ( x  =  y  <->  y  =  x )
64, 5bitri 240 . . . . 5  |-  ( x  _I  y  <->  y  =  x )
76exbii 1572 . . . 4  |-  ( E. y  x  _I  y  <->  E. y  y  =  x )
82, 7mpbir 200 . . 3  |-  E. y  x  _I  y
9 vex 2804 . . . 4  |-  x  e. 
_V
109eldm 4892 . . 3  |-  ( x  e.  dom  _I  <->  E. y  x  _I  y )
118, 10mpbir 200 . 2  |-  x  e. 
dom  _I
121, 11mpgbir 1540 1  |-  dom  _I  =  _V
Colors of variables: wff set class
Syntax hints:   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801   class class class wbr 4039    _I cid 4320   dom cdm 4705
This theorem is referenced by:  dmv  4910  iprc  4959  dmresi  5021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-dm 4715
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