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Theorem ecid 6446
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 2660 . . . 4  |-  y  e. 
_V
2 ecid.1 . . . 4  |-  A  e. 
_V
31, 2elec 6422 . . 3  |-  ( y  e.  [ A ] `'  _E  <->  A `'  _E  y
)
42, 1brcnv 4682 . . 3  |-  ( A `'  _E  y  <->  y  _E  A )
52epelc 4173 . . 3  |-  ( y  _E  A  <->  y  e.  A )
63, 4, 53bitri 205 . 2  |-  ( y  e.  [ A ] `'  _E  <->  y  e.  A
)
76eqriv 2112 1  |-  [ A ] `'  _E  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1314    e. wcel 1463   _Vcvv 2657   class class class wbr 3895    _E cep 4169   `'ccnv 4498   [cec 6381
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 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 4006  ax-pow 4058  ax-pr 4091
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 2244  df-ral 2395  df-rex 2396  df-v 2659  df-sbc 2879  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-opab 3950  df-eprel 4171  df-xp 4505  df-cnv 4507  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-ec 6385
This theorem is referenced by:  qsid  6448
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