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Mirrors > Home > ILE Home > Th. List > resdm | GIF version |
Description: A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.) |
Ref | Expression |
---|---|
resdm | ⊢ (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3044 | . 2 ⊢ dom 𝐴 ⊆ dom 𝐴 | |
2 | relssres 4750 | . 2 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ dom 𝐴) → (𝐴 ↾ dom 𝐴) = 𝐴) | |
3 | 1, 2 | mpan2 416 | 1 ⊢ (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1289 ⊆ wss 2999 dom cdm 4438 ↾ 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-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-dm 4448 df-res 4450 |
This theorem is referenced by: resindm 4754 resdm2 4921 relresfld 4960 relcoi1 4962 funimaexg 5098 fnex 5519 dftpos2 6026 pmresg 6433 dif1en 6595 |
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