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Theorem ecidg 6106
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 (A 𝑉 → [A] E = A)

Proof of Theorem ecidg
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 vex 2554 . . . 4 y V
2 elecg 6080 . . . 4 ((y V A 𝑉) → (y [A] E ↔ A E y))
31, 2mpan 400 . . 3 (A 𝑉 → (y [A] E ↔ A E y))
4 brcnvg 4459 . . . 4 ((A 𝑉 y V) → (A E yy E A))
51, 4mpan2 401 . . 3 (A 𝑉 → (A E yy E A))
6 epelg 4018 . . 3 (A 𝑉 → (y E Ay A))
73, 5, 63bitrd 203 . 2 (A 𝑉 → (y [A] E ↔ y A))
87eqrdv 2035 1 (A 𝑉 → [A] E = A)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242   wcel 1390  Vcvv 2551   class class class wbr 3755   E cep 4015  ccnv 4287  [cec 6040
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-eprel 4017  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-ec 6044
This theorem is referenced by:  addcnsrec  6739  mulcnsrec  6740
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