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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 5006 | . . 3 ⊢ Rel ◡◡𝑅 | |
2 | vex 2740 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 2740 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | opelcnv 4809 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ ◡◡𝑅 ↔ 〈𝑦, 𝑥〉 ∈ ◡𝑅) |
5 | 3, 2 | opelcnv 4809 | . . . . 5 ⊢ (〈𝑦, 𝑥〉 ∈ ◡𝑅 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) |
6 | 4, 5 | bitri 184 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ ◡◡𝑅 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) |
7 | 6 | eqrelriv 4719 | . . 3 ⊢ ((Rel ◡◡𝑅 ∧ Rel 𝑅) → ◡◡𝑅 = 𝑅) |
8 | 1, 7 | mpan 424 | . 2 ⊢ (Rel 𝑅 → ◡◡𝑅 = 𝑅) |
9 | releq 4708 | . . 3 ⊢ (◡◡𝑅 = 𝑅 → (Rel ◡◡𝑅 ↔ Rel 𝑅)) | |
10 | 1, 9 | mpbii 148 | . 2 ⊢ (◡◡𝑅 = 𝑅 → Rel 𝑅) |
11 | 8, 10 | impbii 126 | 1 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 = wceq 1353 ∈ wcel 2148 〈cop 3595 ◡ccnv 4625 Rel wrel 4631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-br 4004 df-opab 4065 df-xp 4632 df-rel 4633 df-cnv 4634 |
This theorem is referenced by: dfrel4v 5080 cnvcnv 5081 cnveqb 5084 dfrel3 5086 cnvcnvres 5092 cnvsn 5111 cores2 5141 co01 5143 coi2 5145 relcnvtr 5148 relcnvexb 5168 funcnvres2 5291 f1cnvcnv 5432 f1ocnv 5474 f1ocnvb 5475 f1ococnv1 5490 isores1 5814 cnvf1o 6225 tposf12 6269 ssenen 6850 relcnvfi 6939 caseinl 7089 caseinr 7090 fsumcnv 11444 fprodcnv 11632 structcnvcnv 12477 hmeocnv 13743 hmeocnvb 13754 |
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