Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ecid Structured version   Visualization version   GIF version

Theorem ecid 7699
 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 3176 . . . 4 𝑦 ∈ V
2 ecid.1 . . . 4 𝐴 ∈ V
31, 2elec 7673 . . 3 (𝑦 ∈ [𝐴] E ↔ 𝐴 E 𝑦)
42, 1brcnv 5227 . . 3 (𝐴 E 𝑦𝑦 E 𝐴)
52epelc 4951 . . 3 (𝑦 E 𝐴𝑦𝐴)
63, 4, 53bitri 285 . 2 (𝑦 ∈ [𝐴] E ↔ 𝑦𝐴)
76eqriv 2607 1 [𝐴] E = 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583   E cep 4947  ◡ccnv 5037  [cec 7627 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-eprel 4949  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ec 7631 This theorem is referenced by:  qsid  7700  addcnsrec  9843  mulcnsrec  9844
 Copyright terms: Public domain W3C validator