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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 4887 | . . 3 ⊢ Rel ◡◡𝑅 | |
2 | vex 2663 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 2663 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | opelcnv 4691 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ ◡◡𝑅 ↔ 〈𝑦, 𝑥〉 ∈ ◡𝑅) |
5 | 3, 2 | opelcnv 4691 | . . . . 5 ⊢ (〈𝑦, 𝑥〉 ∈ ◡𝑅 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) |
6 | 4, 5 | bitri 183 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ ◡◡𝑅 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) |
7 | 6 | eqrelriv 4602 | . . 3 ⊢ ((Rel ◡◡𝑅 ∧ Rel 𝑅) → ◡◡𝑅 = 𝑅) |
8 | 1, 7 | mpan 420 | . 2 ⊢ (Rel 𝑅 → ◡◡𝑅 = 𝑅) |
9 | releq 4591 | . . 3 ⊢ (◡◡𝑅 = 𝑅 → (Rel ◡◡𝑅 ↔ Rel 𝑅)) | |
10 | 1, 9 | mpbii 147 | . 2 ⊢ (◡◡𝑅 = 𝑅 → Rel 𝑅) |
11 | 8, 10 | impbii 125 | 1 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1316 ∈ wcel 1465 〈cop 3500 ◡ccnv 4508 Rel wrel 4514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-xp 4515 df-rel 4516 df-cnv 4517 |
This theorem is referenced by: dfrel4v 4960 cnvcnv 4961 cnveqb 4964 dfrel3 4966 cnvcnvres 4972 cnvsn 4991 cores2 5021 co01 5023 coi2 5025 relcnvtr 5028 relcnvexb 5048 funcnvres2 5168 f1cnvcnv 5309 f1ocnv 5348 f1ocnvb 5349 f1ococnv1 5364 isores1 5683 cnvf1o 6090 tposf12 6134 ssenen 6713 relcnvfi 6797 caseinl 6944 caseinr 6945 fsumcnv 11174 structcnvcnv 11902 hmeocnv 12403 hmeocnvb 12414 |
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