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Theorem ecidg 6200
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 2577 . . . 4 𝑦 ∈ V
2 elecg 6174 . . . 4 ((𝑦 ∈ V ∧ 𝐴𝑉) → (𝑦 ∈ [𝐴] E ↔ 𝐴 E 𝑦))
31, 2mpan 408 . . 3 (𝐴𝑉 → (𝑦 ∈ [𝐴] E ↔ 𝐴 E 𝑦))
4 brcnvg 4543 . . . 4 ((𝐴𝑉𝑦 ∈ V) → (𝐴 E 𝑦𝑦 E 𝐴))
51, 4mpan2 409 . . 3 (𝐴𝑉 → (𝐴 E 𝑦𝑦 E 𝐴))
6 epelg 4054 . . 3 (𝐴𝑉 → (𝑦 E 𝐴𝑦𝐴))
73, 5, 63bitrd 207 . 2 (𝐴𝑉 → (𝑦 ∈ [𝐴] E ↔ 𝑦𝐴))
87eqrdv 2054 1 (𝐴𝑉 → [𝐴] E = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102   = wceq 1259  wcel 1409  Vcvv 2574   class class class wbr 3791   E cep 4051  ccnv 4371  [cec 6134
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-eprel 4053  df-xp 4378  df-cnv 4380  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-ec 6138
This theorem is referenced by:  addcnsrec  6975  mulcnsrec  6976
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