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Theorem dmi 5026
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 3588 . 2  |-  ( dom 
_I  =  _V  <->  A. x  x  e.  dom  _I  )
2 a9ev 1663 . . . 4  |-  E. y 
y  =  x
3 vex 2904 . . . . . . 7  |-  y  e. 
_V
43ideq 4967 . . . . . 6  |-  ( x  _I  y  <->  x  =  y )
5 equcom 1687 . . . . . 6  |-  ( x  =  y  <->  y  =  x )
64, 5bitri 241 . . . . 5  |-  ( x  _I  y  <->  y  =  x )
76exbii 1589 . . . 4  |-  ( E. y  x  _I  y  <->  E. y  y  =  x )
82, 7mpbir 201 . . 3  |-  E. y  x  _I  y
9 vex 2904 . . . 4  |-  x  e. 
_V
109eldm 5009 . . 3  |-  ( x  e.  dom  _I  <->  E. y  x  _I  y )
118, 10mpbir 201 . 2  |-  x  e. 
dom  _I
121, 11mpgbir 1556 1  |-  dom  _I  =  _V
Colors of variables: wff set class
Syntax hints:   E.wex 1547    = wceq 1649    e. wcel 1717   _Vcvv 2901   class class class wbr 4155    _I cid 4436   dom cdm 4820
This theorem is referenced by:  dmv  5027  iprc  5076  dmresi  5138
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-br 4156  df-opab 4210  df-id 4441  df-xp 4826  df-rel 4827  df-dm 4830
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