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Mirrors > Home > ILE Home > Th. List > dfrel2 | GIF version |
Description: Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 29-Dec-1996.) |
Ref | Expression |
---|---|
dfrel2 | ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 4723 | . . 3 ⊢ Rel ◡◡𝑅 | |
2 | vex 2604 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 2604 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | opelcnv 4535 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ ◡◡𝑅 ↔ 〈𝑦, 𝑥〉 ∈ ◡𝑅) |
5 | 3, 2 | opelcnv 4535 | . . . . 5 ⊢ (〈𝑦, 𝑥〉 ∈ ◡𝑅 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) |
6 | 4, 5 | bitri 182 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ ◡◡𝑅 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) |
7 | 6 | eqrelriv 4451 | . . 3 ⊢ ((Rel ◡◡𝑅 ∧ Rel 𝑅) → ◡◡𝑅 = 𝑅) |
8 | 1, 7 | mpan 414 | . 2 ⊢ (Rel 𝑅 → ◡◡𝑅 = 𝑅) |
9 | releq 4440 | . . 3 ⊢ (◡◡𝑅 = 𝑅 → (Rel ◡◡𝑅 ↔ Rel 𝑅)) | |
10 | 1, 9 | mpbii 146 | . 2 ⊢ (◡◡𝑅 = 𝑅 → Rel 𝑅) |
11 | 8, 10 | impbii 124 | 1 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 = wceq 1284 ∈ wcel 1433 〈cop 3401 ◡ccnv 4362 Rel wrel 4368 |
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 |
This theorem is referenced by: dfrel4v 4792 cnvcnv 4793 cnveqb 4796 dfrel3 4798 cnvcnvres 4804 cnvsn 4823 cores2 4853 co01 4855 coi2 4857 relcnvtr 4860 relcnvexb 4877 funcnvres2 4994 f1cnvcnv 5120 f1ocnv 5159 f1ocnvb 5160 f1ococnv1 5175 isores1 5474 cnvf1o 5866 tposf12 5907 relcnvfi 6391 |
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