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 5110 | . 2 ⊢ (𝑆 ∘ 𝑅) ⊆ (dom 𝑅 × ran 𝑆) | |
2 | cossxp2.r | . . . 4 ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵)) | |
3 | dmxpss2 5020 | . . . 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 5021 | . . . 4 ⊢ (𝑆 ⊆ (𝐵 × 𝐶) → ran 𝑆 ⊆ 𝐶) | |
7 | 5, 6 | syl 14 | . . 3 ⊢ (𝜑 → ran 𝑆 ⊆ 𝐶) |
8 | xpss12 4695 | . . 3 ⊢ ((dom 𝑅 ⊆ 𝐴 ∧ ran 𝑆 ⊆ 𝐶) → (dom 𝑅 × ran 𝑆) ⊆ (𝐴 × 𝐶)) | |
9 | 4, 7, 8 | syl2anc 409 | . 2 ⊢ (𝜑 → (dom 𝑅 × ran 𝑆) ⊆ (𝐴 × 𝐶)) |
10 | 1, 9 | sstrid 3139 | 1 ⊢ (𝜑 → (𝑆 ∘ 𝑅) ⊆ (𝐴 × 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ⊆ wss 3102 × cxp 4586 dom cdm 4588 ran crn 4589 ∘ ccom 4592 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-br 3968 df-opab 4028 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |