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Theorem dmi 5076
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 3635 . 2  |-  ( dom 
_I  =  _V  <->  A. x  x  e.  dom  _I  )
2 a9ev 1668 . . . 4  |-  E. y 
y  =  x
3 vex 2951 . . . . . . 7  |-  y  e. 
_V
43ideq 5017 . . . . . 6  |-  ( x  _I  y  <->  x  =  y )
5 equcom 1692 . . . . . 6  |-  ( x  =  y  <->  y  =  x )
64, 5bitri 241 . . . . 5  |-  ( x  _I  y  <->  y  =  x )
76exbii 1592 . . . 4  |-  ( E. y  x  _I  y  <->  E. y  y  =  x )
82, 7mpbir 201 . . 3  |-  E. y  x  _I  y
9 vex 2951 . . . 4  |-  x  e. 
_V
109eldm 5059 . . 3  |-  ( x  e.  dom  _I  <->  E. y  x  _I  y )
118, 10mpbir 201 . 2  |-  x  e. 
dom  _I
121, 11mpgbir 1559 1  |-  dom  _I  =  _V
Colors of variables: wff set class
Syntax hints:   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2948   class class class wbr 4204    _I cid 4485   dom cdm 4870
This theorem is referenced by:  dmv  5077  iprc  5126  dmresi  5188
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-dm 4880
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