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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 5133 | . 2 ⊢ (𝑆 ∘ 𝑅) ⊆ (dom 𝑅 × ran 𝑆) | |
2 | cossxp2.r | . . . 4 ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵)) | |
3 | dmxpss2 5043 | . . . 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 5044 | . . . 4 ⊢ (𝑆 ⊆ (𝐵 × 𝐶) → ran 𝑆 ⊆ 𝐶) | |
7 | 5, 6 | syl 14 | . . 3 ⊢ (𝜑 → ran 𝑆 ⊆ 𝐶) |
8 | xpss12 4718 | . . 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 3158 | 1 ⊢ (𝜑 → (𝑆 ∘ 𝑅) ⊆ (𝐴 × 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ⊆ wss 3121 × cxp 4609 dom cdm 4611 ran crn 4612 ∘ ccom 4615 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 |
This theorem is referenced by: (None) |
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