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Theorem ecidg 6544
Description: A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by Jim Kingdon, 8-Jan-2020.)
Assertion
Ref Expression
ecidg  |-  ( A  e.  V  ->  [ A ] `'  _E  =  A )

Proof of Theorem ecidg
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 vex 2715 . . . 4  |-  y  e. 
_V
2 elecg 6518 . . . 4  |-  ( ( y  e.  _V  /\  A  e.  V )  ->  ( y  e.  [ A ] `'  _E  <->  A `'  _E  y ) )
31, 2mpan 421 . . 3  |-  ( A  e.  V  ->  (
y  e.  [ A ] `'  _E  <->  A `'  _E  y ) )
4 brcnvg 4767 . . . 4  |-  ( ( A  e.  V  /\  y  e.  _V )  ->  ( A `'  _E  y 
<->  y  _E  A ) )
51, 4mpan2 422 . . 3  |-  ( A  e.  V  ->  ( A `'  _E  y  <->  y  _E  A ) )
6 epelg 4250 . . 3  |-  ( A  e.  V  ->  (
y  _E  A  <->  y  e.  A ) )
73, 5, 63bitrd 213 . 2  |-  ( A  e.  V  ->  (
y  e.  [ A ] `'  _E  <->  y  e.  A ) )
87eqrdv 2155 1  |-  ( A  e.  V  ->  [ A ] `'  _E  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1335    e. wcel 2128   _Vcvv 2712   class class class wbr 3965    _E cep 4247   `'ccnv 4585   [cec 6478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-eprel 4249  df-xp 4592  df-cnv 4594  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-ec 6482
This theorem is referenced by:  addcnsrec  7762  mulcnsrec  7763
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