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Theorem ercnv 7723
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 7711 . 2 (𝑅 Er 𝐴 → Rel 𝑅)
2 relcnv 5472 . . 3 Rel 𝑅
3 id 22 . . . . . 6 (𝑅 Er 𝐴𝑅 Er 𝐴)
43ersymb 7716 . . . . 5 (𝑅 Er 𝐴 → (𝑦𝑅𝑥𝑥𝑅𝑦))
5 vex 3193 . . . . . . 7 𝑥 ∈ V
6 vex 3193 . . . . . . 7 𝑦 ∈ V
75, 6brcnv 5275 . . . . . 6 (𝑥𝑅𝑦𝑦𝑅𝑥)
8 df-br 4624 . . . . . 6 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
97, 8bitr3i 266 . . . . 5 (𝑦𝑅𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
10 df-br 4624 . . . . 5 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
114, 9, 103bitr3g 302 . . . 4 (𝑅 Er 𝐴 → (⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
1211eqrelrdv2 5190 . . 3 (((Rel 𝑅 ∧ Rel 𝑅) ∧ 𝑅 Er 𝐴) → 𝑅 = 𝑅)
132, 12mpanl1 715 . 2 ((Rel 𝑅𝑅 Er 𝐴) → 𝑅 = 𝑅)
141, 13mpancom 702 1 (𝑅 Er 𝐴𝑅 = 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  cop 4161   class class class wbr 4623  ccnv 5083  Rel wrel 5089   Er wer 7699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-xp 5090  df-rel 5091  df-cnv 5092  df-er 7702
This theorem is referenced by:  errn  7724
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