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| Description: The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.) | 
| Ref | Expression | 
|---|---|
| ercnv | ⊢ (𝑅 Er 𝐴 → ◡𝑅 = 𝑅) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | errel 8754 | . 2 ⊢ (𝑅 Er 𝐴 → Rel 𝑅) | |
| 2 | relcnv 6122 | . . 3 ⊢ Rel ◡𝑅 | |
| 3 | id 22 | . . . . . 6 ⊢ (𝑅 Er 𝐴 → 𝑅 Er 𝐴) | |
| 4 | 3 | ersymb 8759 | . . . . 5 ⊢ (𝑅 Er 𝐴 → (𝑦𝑅𝑥 ↔ 𝑥𝑅𝑦)) | 
| 5 | vex 3484 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
| 6 | vex 3484 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
| 7 | 5, 6 | brcnv 5893 | . . . . . 6 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥) | 
| 8 | df-br 5144 | . . . . . 6 ⊢ (𝑥◡𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ ◡𝑅) | |
| 9 | 7, 8 | bitr3i 277 | . . . . 5 ⊢ (𝑦𝑅𝑥 ↔ 〈𝑥, 𝑦〉 ∈ ◡𝑅) | 
| 10 | df-br 5144 | . . . . 5 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) | |
| 11 | 4, 9, 10 | 3bitr3g 313 | . . . 4 ⊢ (𝑅 Er 𝐴 → (〈𝑥, 𝑦〉 ∈ ◡𝑅 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅)) | 
| 12 | 11 | eqrelrdv2 5805 | . . 3 ⊢ (((Rel ◡𝑅 ∧ Rel 𝑅) ∧ 𝑅 Er 𝐴) → ◡𝑅 = 𝑅) | 
| 13 | 2, 12 | mpanl1 700 | . 2 ⊢ ((Rel 𝑅 ∧ 𝑅 Er 𝐴) → ◡𝑅 = 𝑅) | 
| 14 | 1, 13 | mpancom 688 | 1 ⊢ (𝑅 Er 𝐴 → ◡𝑅 = 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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