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Theorem ecid 6724
Description: A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
Hypothesis
Ref Expression
ecid.1  |-  A  e. 
_V
Assertion
Ref Expression
ecid  |-  [ A ] `'  _E  =  A

Proof of Theorem ecid
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 vex 2791 . . . 4  |-  y  e. 
_V
2 ecid.1 . . . 4  |-  A  e. 
_V
31, 2elec 6699 . . 3  |-  ( y  e.  [ A ] `'  _E  <->  A `'  _E  y
)
42, 1brcnv 4864 . . 3  |-  ( A `'  _E  y  <->  y  _E  A )
52epelc 4307 . . 3  |-  ( y  _E  A  <->  y  e.  A )
63, 4, 53bitri 262 . 2  |-  ( y  e.  [ A ] `'  _E  <->  y  e.  A
)
76eqriv 2280 1  |-  [ A ] `'  _E  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   _Vcvv 2788   class class class wbr 4023    _E cep 4303   `'ccnv 4688   [cec 6658
This theorem is referenced by:  qsid  6725  addcnsrec  8765  mulcnsrec  8766
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-eprel 4305  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-ec 6662
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