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Theorem ecidg 6496
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 (𝐴𝑉 → [𝐴] E = 𝐴)

Proof of Theorem ecidg
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 vex 2689 . . . 4 𝑦 ∈ V
2 elecg 6470 . . . 4 ((𝑦 ∈ V ∧ 𝐴𝑉) → (𝑦 ∈ [𝐴] E ↔ 𝐴 E 𝑦))
31, 2mpan 420 . . 3 (𝐴𝑉 → (𝑦 ∈ [𝐴] E ↔ 𝐴 E 𝑦))
4 brcnvg 4723 . . . 4 ((𝐴𝑉𝑦 ∈ V) → (𝐴 E 𝑦𝑦 E 𝐴))
51, 4mpan2 421 . . 3 (𝐴𝑉 → (𝐴 E 𝑦𝑦 E 𝐴))
6 epelg 4215 . . 3 (𝐴𝑉 → (𝑦 E 𝐴𝑦𝐴))
73, 5, 63bitrd 213 . 2 (𝐴𝑉 → (𝑦 ∈ [𝐴] E ↔ 𝑦𝐴))
87eqrdv 2137 1 (𝐴𝑉 → [𝐴] E = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1331  wcel 1480  Vcvv 2686   class class class wbr 3932   E cep 4212  ccnv 4541  [cec 6430
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049  ax-pow 4101  ax-pr 4134
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3933  df-opab 3993  df-eprel 4214  df-xp 4548  df-cnv 4550  df-dm 4552  df-rn 4553  df-res 4554  df-ima 4555  df-ec 6434
This theorem is referenced by:  addcnsrec  7669  mulcnsrec  7670
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