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Theorem ecid 7439
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 3083 . . . 4  |-  y  e. 
_V
2 ecid.1 . . . 4  |-  A  e. 
_V
31, 2elec 7414 . . 3  |-  ( y  e.  [ A ] `'  _E  <->  A `'  _E  y
)
42, 1brcnv 5036 . . 3  |-  ( A `'  _E  y  <->  y  _E  A )
52epelc 4766 . . 3  |-  ( y  _E  A  <->  y  e.  A )
63, 4, 53bitri 274 . 2  |-  ( y  e.  [ A ] `'  _E  <->  y  e.  A
)
76eqriv 2418 1  |-  [ A ] `'  _E  =  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437    e. wcel 1872   _Vcvv 3080   class class class wbr 4423    _E cep 4762   `'ccnv 4852   [cec 7372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-br 4424  df-opab 4483  df-eprel 4764  df-xp 4859  df-cnv 4861  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-ec 7376
This theorem is referenced by:  qsid  7440  addcnsrec  9574  mulcnsrec  9575
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