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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 4646 | . . 3 ⊢ Rel ◡◡𝑅 | |
2 | vex 2554 | . . . . . 6 ⊢ x ∈ V | |
3 | vex 2554 | . . . . . 6 ⊢ y ∈ V | |
4 | 2, 3 | opelcnv 4460 | . . . . 5 ⊢ (〈x, y〉 ∈ ◡◡𝑅 ↔ 〈y, x〉 ∈ ◡𝑅) |
5 | 3, 2 | opelcnv 4460 | . . . . 5 ⊢ (〈y, x〉 ∈ ◡𝑅 ↔ 〈x, y〉 ∈ 𝑅) |
6 | 4, 5 | bitri 173 | . . . 4 ⊢ (〈x, y〉 ∈ ◡◡𝑅 ↔ 〈x, y〉 ∈ 𝑅) |
7 | 6 | eqrelriv 4376 | . . 3 ⊢ ((Rel ◡◡𝑅 ∧ Rel 𝑅) → ◡◡𝑅 = 𝑅) |
8 | 1, 7 | mpan 400 | . 2 ⊢ (Rel 𝑅 → ◡◡𝑅 = 𝑅) |
9 | releq 4365 | . . 3 ⊢ (◡◡𝑅 = 𝑅 → (Rel ◡◡𝑅 ↔ Rel 𝑅)) | |
10 | 1, 9 | mpbii 136 | . 2 ⊢ (◡◡𝑅 = 𝑅 → Rel 𝑅) |
11 | 8, 10 | impbii 117 | 1 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 = wceq 1242 ∈ wcel 1390 〈cop 3370 ◡ccnv 4287 Rel wrel 4293 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-xp 4294 df-rel 4295 df-cnv 4296 |
This theorem is referenced by: dfrel4v 4715 cnvcnv 4716 cnveqb 4719 dfrel3 4721 cnvcnvres 4727 cnvsn 4746 cores2 4776 co01 4778 coi2 4780 relcnvtr 4783 relcnvexb 4800 funcnvres2 4917 f1cnvcnv 5043 f1ocnv 5082 f1ocnvb 5083 f1ococnv1 5098 isores1 5397 cnvf1o 5788 tposf12 5825 |
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