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Theorem ecidg 6459
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 2661 . . . 4  |-  y  e. 
_V
2 elecg 6433 . . . 4  |-  ( ( y  e.  _V  /\  A  e.  V )  ->  ( y  e.  [ A ] `'  _E  <->  A `'  _E  y ) )
31, 2mpan 418 . . 3  |-  ( A  e.  V  ->  (
y  e.  [ A ] `'  _E  <->  A `'  _E  y ) )
4 brcnvg 4688 . . . 4  |-  ( ( A  e.  V  /\  y  e.  _V )  ->  ( A `'  _E  y 
<->  y  _E  A ) )
51, 4mpan2 419 . . 3  |-  ( A  e.  V  ->  ( A `'  _E  y  <->  y  _E  A ) )
6 epelg 4180 . . 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 2113 1  |-  ( A  e.  V  ->  [ A ] `'  _E  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1314    e. wcel 1463   _Vcvv 2658   class class class wbr 3897    _E cep 4177   `'ccnv 4506   [cec 6393
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-eprel 4179  df-xp 4513  df-cnv 4515  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-ec 6397
This theorem is referenced by:  addcnsrec  7614  mulcnsrec  7615
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