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Theorem ercnv 6293
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 6281 . 2 (𝑅 Er 𝐴 → Rel 𝑅)
2 relcnv 4797 . . 3 Rel 𝑅
3 id 19 . . . . . 6 (𝑅 Er 𝐴𝑅 Er 𝐴)
43ersymb 6286 . . . . 5 (𝑅 Er 𝐴 → (𝑦𝑅𝑥𝑥𝑅𝑦))
5 vex 2622 . . . . . . 7 𝑥 ∈ V
6 vex 2622 . . . . . . 7 𝑦 ∈ V
75, 6brcnv 4607 . . . . . 6 (𝑥𝑅𝑦𝑦𝑅𝑥)
8 df-br 3838 . . . . . 6 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
97, 8bitr3i 184 . . . . 5 (𝑦𝑅𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
10 df-br 3838 . . . . 5 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
114, 9, 103bitr3g 220 . . . 4 (𝑅 Er 𝐴 → (⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
1211eqrelrdv2 4525 . . 3 (((Rel 𝑅 ∧ Rel 𝑅) ∧ 𝑅 Er 𝐴) → 𝑅 = 𝑅)
132, 12mpanl1 425 . 2 ((Rel 𝑅𝑅 Er 𝐴) → 𝑅 = 𝑅)
141, 13mpancom 413 1 (𝑅 Er 𝐴𝑅 = 𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  wcel 1438  cop 3444   class class class wbr 3837  ccnv 4427  Rel wrel 4433   Er wer 6269
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-cnv 4436  df-er 6272
This theorem is referenced by:  errn  6294
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