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Mirrors > Home > ILE Home > Th. List > ercnv | 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 6138 | . 2 ⊢ (𝑅 Er 𝐴 → Rel 𝑅) | |
2 | relcnv 4723 | . . 3 ⊢ Rel ◡𝑅 | |
3 | id 19 | . . . . . 6 ⊢ (𝑅 Er 𝐴 → 𝑅 Er 𝐴) | |
4 | 3 | ersymb 6143 | . . . . 5 ⊢ (𝑅 Er 𝐴 → (𝑦𝑅𝑥 ↔ 𝑥𝑅𝑦)) |
5 | vex 2604 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
6 | vex 2604 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
7 | 5, 6 | brcnv 4536 | . . . . . 6 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥) |
8 | df-br 3786 | . . . . . 6 ⊢ (𝑥◡𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ ◡𝑅) | |
9 | 7, 8 | bitr3i 184 | . . . . 5 ⊢ (𝑦𝑅𝑥 ↔ 〈𝑥, 𝑦〉 ∈ ◡𝑅) |
10 | df-br 3786 | . . . . 5 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) | |
11 | 4, 9, 10 | 3bitr3g 220 | . . . 4 ⊢ (𝑅 Er 𝐴 → (〈𝑥, 𝑦〉 ∈ ◡𝑅 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅)) |
12 | 11 | eqrelrdv2 4457 | . . 3 ⊢ (((Rel ◡𝑅 ∧ Rel 𝑅) ∧ 𝑅 Er 𝐴) → ◡𝑅 = 𝑅) |
13 | 2, 12 | mpanl1 424 | . 2 ⊢ ((Rel 𝑅 ∧ 𝑅 Er 𝐴) → ◡𝑅 = 𝑅) |
14 | 1, 13 | mpancom 413 | 1 ⊢ (𝑅 Er 𝐴 → ◡𝑅 = 𝑅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 ∈ wcel 1433 〈cop 3401 class class class wbr 3785 ◡ccnv 4362 Rel wrel 4368 Er wer 6126 |
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 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-xp 4369 df-rel 4370 df-cnv 4371 df-er 6129 |
This theorem is referenced by: errn 6151 |
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