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Theorem ercnv 8766
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 8754 . 2 (𝑅 Er 𝐴 → Rel 𝑅)
2 relcnv 6122 . . 3 Rel 𝑅
3 id 22 . . . . . 6 (𝑅 Er 𝐴𝑅 Er 𝐴)
43ersymb 8759 . . . . 5 (𝑅 Er 𝐴 → (𝑦𝑅𝑥𝑥𝑅𝑦))
5 vex 3484 . . . . . . 7 𝑥 ∈ V
6 vex 3484 . . . . . . 7 𝑦 ∈ V
75, 6brcnv 5893 . . . . . 6 (𝑥𝑅𝑦𝑦𝑅𝑥)
8 df-br 5144 . . . . . 6 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
97, 8bitr3i 277 . . . . 5 (𝑦𝑅𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
10 df-br 5144 . . . . 5 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
114, 9, 103bitr3g 313 . . . 4 (𝑅 Er 𝐴 → (⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
1211eqrelrdv2 5805 . . 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 1540  wcel 2108  cop 4632   class class class wbr 5143  ccnv 5684  Rel wrel 5690   Er wer 8742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-er 8745
This theorem is referenced by:  errn  8767  prjspeclsp  42622
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