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Theorem ecidg 6201
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 2577 . . . 4  |-  y  e. 
_V
2 elecg 6175 . . . 4  |-  ( ( y  e.  _V  /\  A  e.  V )  ->  ( y  e.  [ A ] `'  _E  <->  A `'  _E  y ) )
31, 2mpan 408 . . 3  |-  ( A  e.  V  ->  (
y  e.  [ A ] `'  _E  <->  A `'  _E  y ) )
4 brcnvg 4544 . . . 4  |-  ( ( A  e.  V  /\  y  e.  _V )  ->  ( A `'  _E  y 
<->  y  _E  A ) )
51, 4mpan2 409 . . 3  |-  ( A  e.  V  ->  ( A `'  _E  y  <->  y  _E  A ) )
6 epelg 4055 . . 3  |-  ( A  e.  V  ->  (
y  _E  A  <->  y  e.  A ) )
73, 5, 63bitrd 207 . 2  |-  ( A  e.  V  ->  (
y  e.  [ A ] `'  _E  <->  y  e.  A ) )
87eqrdv 2054 1  |-  ( A  e.  V  ->  [ A ] `'  _E  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102    = wceq 1259    e. wcel 1409   _Vcvv 2574   class class class wbr 3792    _E cep 4052   `'ccnv 4372   [cec 6135
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-eprel 4054  df-xp 4379  df-cnv 4381  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-ec 6139
This theorem is referenced by:  addcnsrec  6976  mulcnsrec  6977
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