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Theorem ecid 7855
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 ∈ 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 3234 . . . 4 𝑦 ∈ V
2 ecid.1 . . . 4 𝐴 ∈ V
31, 2elec 7829 . . 3 (𝑦 ∈ [𝐴] E ↔ 𝐴 E 𝑦)
42, 1brcnv 5337 . . 3 (𝐴 E 𝑦𝑦 E 𝐴)
52epelc 5060 . . 3 (𝑦 E 𝐴𝑦𝐴)
63, 4, 53bitri 286 . 2 (𝑦 ∈ [𝐴] E ↔ 𝑦𝐴)
76eqriv 2648 1 [𝐴] E = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wcel 2030  Vcvv 3231   class class class wbr 4685   E cep 5057  ccnv 5142  [cec 7785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-eprel 5058  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ec 7789
This theorem is referenced by:  qsid  7856  addcnsrec  10002  mulcnsrec  10003
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