Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > cossxp2 | GIF version |
Description: The composition of two relations is a relation, with bounds on its domain and codomain. (Contributed by BJ, 10-Jul-2022.) |
Ref | Expression |
---|---|
cossxp2.r | ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵)) |
cossxp2.s | ⊢ (𝜑 → 𝑆 ⊆ (𝐵 × 𝐶)) |
Ref | Expression |
---|---|
cossxp2 | ⊢ (𝜑 → (𝑆 ∘ 𝑅) ⊆ (𝐴 × 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cossxp 5061 | . 2 ⊢ (𝑆 ∘ 𝑅) ⊆ (dom 𝑅 × ran 𝑆) | |
2 | cossxp2.r | . . . 4 ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵)) | |
3 | dmxpss2 4971 | . . . 4 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → dom 𝑅 ⊆ 𝐴) | |
4 | 2, 3 | syl 14 | . . 3 ⊢ (𝜑 → dom 𝑅 ⊆ 𝐴) |
5 | cossxp2.s | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ (𝐵 × 𝐶)) | |
6 | rnxpss2 4972 | . . . 4 ⊢ (𝑆 ⊆ (𝐵 × 𝐶) → ran 𝑆 ⊆ 𝐶) | |
7 | 5, 6 | syl 14 | . . 3 ⊢ (𝜑 → ran 𝑆 ⊆ 𝐶) |
8 | xpss12 4646 | . . 3 ⊢ ((dom 𝑅 ⊆ 𝐴 ∧ ran 𝑆 ⊆ 𝐶) → (dom 𝑅 × ran 𝑆) ⊆ (𝐴 × 𝐶)) | |
9 | 4, 7, 8 | syl2anc 408 | . 2 ⊢ (𝜑 → (dom 𝑅 × ran 𝑆) ⊆ (𝐴 × 𝐶)) |
10 | 1, 9 | sstrid 3108 | 1 ⊢ (𝜑 → (𝑆 ∘ 𝑅) ⊆ (𝐴 × 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ⊆ wss 3071 × cxp 4537 dom cdm 4539 ran crn 4540 ∘ ccom 4543 |
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-eu 2002 df-mo 2003 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-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |