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Mirrors > Home > ILE Home > Th. List > dfrel3 | GIF version |
Description: Alternate definition of relation. (Contributed by NM, 14-May-2008.) |
Ref | Expression |
---|---|
dfrel3 | ⊢ (Rel 𝑅 ↔ (𝑅 ↾ V) = 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrel2 4881 | . 2 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) | |
2 | cnvcnv2 4884 | . . 3 ⊢ ◡◡𝑅 = (𝑅 ↾ V) | |
3 | 2 | eqeq1i 2095 | . 2 ⊢ (◡◡𝑅 = 𝑅 ↔ (𝑅 ↾ V) = 𝑅) |
4 | 1, 3 | bitri 182 | 1 ⊢ (Rel 𝑅 ↔ (𝑅 ↾ V) = 𝑅) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 = wceq 1289 Vcvv 2619 ◡ccnv 4437 ↾ cres 4440 Rel wrel 4443 |
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 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-br 3846 df-opab 3900 df-xp 4444 df-rel 4445 df-cnv 4446 df-res 4450 |
This theorem is referenced by: cocnvcnv2 4942 f1ovi 5292 |
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