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Theorem dmi 5055
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 3739 . 2  |-  ( dom 
_I  =  _V  <->  A. x  x  e.  dom  _I  )
2 ax6ev 1815 . . . 4  |-  E. y 
y  =  x
3 vex 3034 . . . . . . 7  |-  y  e. 
_V
43ideq 4992 . . . . . 6  |-  ( x  _I  y  <->  x  =  y )
5 equcom 1870 . . . . . 6  |-  ( x  =  y  <->  y  =  x )
64, 5bitri 257 . . . . 5  |-  ( x  _I  y  <->  y  =  x )
76exbii 1726 . . . 4  |-  ( E. y  x  _I  y  <->  E. y  y  =  x )
82, 7mpbir 214 . . 3  |-  E. y  x  _I  y
9 vex 3034 . . . 4  |-  x  e. 
_V
109eldm 5037 . . 3  |-  ( x  e.  dom  _I  <->  E. y  x  _I  y )
118, 10mpbir 214 . 2  |-  x  e. 
dom  _I
121, 11mpgbir 1681 1  |-  dom  _I  =  _V
Colors of variables: wff setvar class
Syntax hints:    = wceq 1452   E.wex 1671    e. wcel 1904   _Vcvv 3031   class class class wbr 4395    _I cid 4749   dom cdm 4839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-dm 4849
This theorem is referenced by:  dmv  5056  dmresi  5166  iprc  6747
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