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Theorem ercnv 8765
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 8753 . 2 (𝑅 Er 𝐴 → Rel 𝑅)
2 relcnv 6125 . . 3 Rel 𝑅
3 id 22 . . . . . 6 (𝑅 Er 𝐴𝑅 Er 𝐴)
43ersymb 8758 . . . . 5 (𝑅 Er 𝐴 → (𝑦𝑅𝑥𝑥𝑅𝑦))
5 vex 3482 . . . . . . 7 𝑥 ∈ V
6 vex 3482 . . . . . . 7 𝑦 ∈ V
75, 6brcnv 5896 . . . . . 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 5808 . . 3 (((Rel 𝑅 ∧ Rel 𝑅) ∧ 𝑅 Er 𝐴) → 𝑅 = 𝑅)
132, 12mpanl1 700 . 2 ((Rel 𝑅𝑅 Er 𝐴) → 𝑅 = 𝑅)
141, 13mpancom 688 1 (𝑅 Er 𝐴𝑅 = 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cop 4637   class class class wbr 5148  ccnv 5688  Rel wrel 5694   Er wer 8741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-er 8744
This theorem is referenced by:  errn  8766  prjspeclsp  42599
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