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Theorem ecid 6898
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 2895 . . . 4  |-  y  e. 
_V
2 ecid.1 . . . 4  |-  A  e. 
_V
31, 2elec 6873 . . 3  |-  ( y  e.  [ A ] `'  _E  <->  A `'  _E  y
)
42, 1brcnv 4988 . . 3  |-  ( A `'  _E  y  <->  y  _E  A )
52epelc 4430 . . 3  |-  ( y  _E  A  <->  y  e.  A )
63, 4, 53bitri 263 . 2  |-  ( y  e.  [ A ] `'  _E  <->  y  e.  A
)
76eqriv 2377 1  |-  [ A ] `'  _E  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1717   _Vcvv 2892   class class class wbr 4146    _E cep 4426   `'ccnv 4810   [cec 6832
This theorem is referenced by:  qsid  6899  addcnsrec  8944  mulcnsrec  8945
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 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-eprel 4428  df-xp 4817  df-cnv 4819  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-ec 6836
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