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Theorem ecid 6615
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 2754 . . . 4  |-  y  e. 
_V
2 ecid.1 . . . 4  |-  A  e. 
_V
31, 2elec 6591 . . 3  |-  ( y  e.  [ A ] `'  _E  <->  A `'  _E  y
)
42, 1brcnv 4824 . . 3  |-  ( A `'  _E  y  <->  y  _E  A )
52epelc 4305 . . 3  |-  ( y  _E  A  <->  y  e.  A )
63, 4, 53bitri 206 . 2  |-  ( y  e.  [ A ] `'  _E  <->  y  e.  A
)
76eqriv 2185 1  |-  [ A ] `'  _E  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1363    e. wcel 2159   _Vcvv 2751   class class class wbr 4017    _E cep 4301   `'ccnv 4639   [cec 6550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rex 2473  df-v 2753  df-sbc 2977  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-br 4018  df-opab 4079  df-eprel 4303  df-xp 4646  df-cnv 4648  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-ec 6554
This theorem is referenced by:  qsid  6617
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