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Mirrors > Home > ILE Home > Th. List > resindm | GIF version |
Description: When restricting a relation, intersecting with the domain of the relation has no effect. (Contributed by FL, 6-Oct-2008.) |
Ref | Expression |
---|---|
resindm | ⊢ (Rel 𝐴 → (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resdm 4858 | . . 3 ⊢ (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴) | |
2 | 1 | ineq2d 3277 | . 2 ⊢ (Rel 𝐴 → ((𝐴 ↾ 𝐵) ∩ (𝐴 ↾ dom 𝐴)) = ((𝐴 ↾ 𝐵) ∩ 𝐴)) |
3 | resindi 4834 | . 2 ⊢ (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = ((𝐴 ↾ 𝐵) ∩ (𝐴 ↾ dom 𝐴)) | |
4 | incom 3268 | . . 3 ⊢ ((𝐴 ↾ 𝐵) ∩ 𝐴) = (𝐴 ∩ (𝐴 ↾ 𝐵)) | |
5 | inres 4836 | . . 3 ⊢ (𝐴 ∩ (𝐴 ↾ 𝐵)) = ((𝐴 ∩ 𝐴) ↾ 𝐵) | |
6 | inidm 3285 | . . . 4 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
7 | 6 | reseq1i 4815 | . . 3 ⊢ ((𝐴 ∩ 𝐴) ↾ 𝐵) = (𝐴 ↾ 𝐵) |
8 | 4, 5, 7 | 3eqtrri 2165 | . 2 ⊢ (𝐴 ↾ 𝐵) = ((𝐴 ↾ 𝐵) ∩ 𝐴) |
9 | 2, 3, 8 | 3eqtr4g 2197 | 1 ⊢ (Rel 𝐴 → (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∩ cin 3070 dom cdm 4539 ↾ cres 4541 Rel wrel 4544 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-dm 4549 df-res 4551 |
This theorem is referenced by: resdmdfsn 4862 |
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