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Theorem domep 30226
Description: The domain of the epsilon relation is the universe. (Contributed by Scott Fenton, 27-Oct-2010.)
Assertion
Ref Expression
domep  |-  dom  _E  =  _V

Proof of Theorem domep
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 equid 1842 . . . 4  |-  x  =  x
2 el 4607 . . . . 5  |-  E. y  x  e.  y
3 epel 4768 . . . . . 6  |-  ( x  _E  y  <->  x  e.  y )
43exbii 1714 . . . . 5  |-  ( E. y  x  _E  y  <->  E. y  x  e.  y )
52, 4mpbir 212 . . . 4  |-  E. y  x  _E  y
61, 52th 242 . . 3  |-  ( x  =  x  <->  E. y  x  _E  y )
76abbii 2563 . 2  |-  { x  |  x  =  x }  =  { x  |  E. y  x  _E  y }
8 df-v 3089 . 2  |-  _V  =  { x  |  x  =  x }
9 df-dm 4864 . 2  |-  dom  _E  =  { x  |  E. y  x  _E  y }
107, 8, 93eqtr4ri 2469 1  |-  dom  _E  =  _V
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437   E.wex 1659   {cab 2414   _Vcvv 3087   class class class wbr 4426    _E cep 4763   dom cdm 4854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-eprel 4765  df-dm 4864
This theorem is referenced by: (None)
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