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

Theorem ercnv 8745
Description: The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.)
Assertion
Ref Expression
ercnv (𝑅 Er 𝐴𝑅 = 𝑅)

Proof of Theorem ercnv
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 errel 8733 . 2 (𝑅 Er 𝐴 → Rel 𝑅)
2 relcnv 6108 . . 3 Rel 𝑅
3 id 22 . . . . . 6 (𝑅 Er 𝐴𝑅 Er 𝐴)
43ersymb 8738 . . . . 5 (𝑅 Er 𝐴 → (𝑦𝑅𝑥𝑥𝑅𝑦))
5 vex 3475 . . . . . . 7 𝑥 ∈ V
6 vex 3475 . . . . . . 7 𝑦 ∈ V
75, 6brcnv 5885 . . . . . 6 (𝑥𝑅𝑦𝑦𝑅𝑥)
8 df-br 5149 . . . . . 6 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
97, 8bitr3i 277 . . . . 5 (𝑦𝑅𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
10 df-br 5149 . . . . 5 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
114, 9, 103bitr3g 313 . . . 4 (𝑅 Er 𝐴 → (⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
1211eqrelrdv2 5797 . . 3 (((Rel 𝑅 ∧ Rel 𝑅) ∧ 𝑅 Er 𝐴) → 𝑅 = 𝑅)
132, 12mpanl1 699 . 2 ((Rel 𝑅𝑅 Er 𝐴) → 𝑅 = 𝑅)
141, 13mpancom 687 1 (𝑅 Er 𝐴𝑅 = 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  cop 4635   class class class wbr 5148  ccnv 5677  Rel wrel 5683   Er wer 8721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-rel 5685  df-cnv 5686  df-er 8724
This theorem is referenced by:  errn  8746  prjspeclsp  42036
  Copyright terms: Public domain W3C validator