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Theorem ecidg 6170
 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 2560 . . . 4 𝑦 ∈ V
2 elecg 6144 . . . 4 ((𝑦 ∈ V ∧ 𝐴𝑉) → (𝑦 ∈ [𝐴] E ↔ 𝐴 E 𝑦))
31, 2mpan 400 . . 3 (𝐴𝑉 → (𝑦 ∈ [𝐴] E ↔ 𝐴 E 𝑦))
4 brcnvg 4516 . . . 4 ((𝐴𝑉𝑦 ∈ V) → (𝐴 E 𝑦𝑦 E 𝐴))
51, 4mpan2 401 . . 3 (𝐴𝑉 → (𝐴 E 𝑦𝑦 E 𝐴))
6 epelg 4027 . . 3 (𝐴𝑉 → (𝑦 E 𝐴𝑦𝐴))
73, 5, 63bitrd 203 . 2 (𝐴𝑉 → (𝑦 ∈ [𝐴] E ↔ 𝑦𝐴))
87eqrdv 2038 1 (𝐴𝑉 → [𝐴] E = 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1243   ∈ wcel 1393  Vcvv 2557   class class class wbr 3764   E cep 4024  ◡ccnv 4344  [cec 6104 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-eprel 4026  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-ec 6108 This theorem is referenced by:  addcnsrec  6918  mulcnsrec  6919
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