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Mirrors > Home > MPE Home > Th. List > ercnv | Structured version Visualization version GIF version |
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 8753 | . 2 ⊢ (𝑅 Er 𝐴 → Rel 𝑅) | |
2 | relcnv 6125 | . . 3 ⊢ Rel ◡𝑅 | |
3 | id 22 | . . . . . 6 ⊢ (𝑅 Er 𝐴 → 𝑅 Er 𝐴) | |
4 | 3 | ersymb 8758 | . . . . 5 ⊢ (𝑅 Er 𝐴 → (𝑦𝑅𝑥 ↔ 𝑥𝑅𝑦)) |
5 | vex 3482 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
6 | vex 3482 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
7 | 5, 6 | brcnv 5896 | . . . . . 6 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥) |
8 | df-br 5149 | . . . . . 6 ⊢ (𝑥◡𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ ◡𝑅) | |
9 | 7, 8 | bitr3i 277 | . . . . 5 ⊢ (𝑦𝑅𝑥 ↔ 〈𝑥, 𝑦〉 ∈ ◡𝑅) |
10 | df-br 5149 | . . . . 5 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) | |
11 | 4, 9, 10 | 3bitr3g 313 | . . . 4 ⊢ (𝑅 Er 𝐴 → (〈𝑥, 𝑦〉 ∈ ◡𝑅 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅)) |
12 | 11 | eqrelrdv2 5808 | . . 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 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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