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Theorem ecidg 6577
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 2733 . . . 4  |-  y  e. 
_V
2 elecg 6551 . . . 4  |-  ( ( y  e.  _V  /\  A  e.  V )  ->  ( y  e.  [ A ] `'  _E  <->  A `'  _E  y ) )
31, 2mpan 422 . . 3  |-  ( A  e.  V  ->  (
y  e.  [ A ] `'  _E  <->  A `'  _E  y ) )
4 brcnvg 4792 . . . 4  |-  ( ( A  e.  V  /\  y  e.  _V )  ->  ( A `'  _E  y 
<->  y  _E  A ) )
51, 4mpan2 423 . . 3  |-  ( A  e.  V  ->  ( A `'  _E  y  <->  y  _E  A ) )
6 epelg 4275 . . 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 2168 1  |-  ( A  e.  V  ->  [ A ] `'  _E  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1348    e. wcel 2141   _Vcvv 2730   class class class wbr 3989    _E cep 4272   `'ccnv 4610   [cec 6511
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-eprel 4274  df-xp 4617  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-ec 6515
This theorem is referenced by:  addcnsrec  7804  mulcnsrec  7805
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