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Mirrors > Home > ILE Home > Th. List > cossxp | GIF version |
Description: Composition as a subset of the cross product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.) |
Ref | Expression |
---|---|
cossxp | ⊢ (𝐴 ∘ 𝐵) ⊆ (dom 𝐵 × ran 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relco 4839 | . . 3 ⊢ Rel (𝐴 ∘ 𝐵) | |
2 | relssdmrn 4861 | . . 3 ⊢ (Rel (𝐴 ∘ 𝐵) → (𝐴 ∘ 𝐵) ⊆ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵))) | |
3 | 1, 2 | ax-mp 7 | . 2 ⊢ (𝐴 ∘ 𝐵) ⊆ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) |
4 | dmcoss 4619 | . . 3 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵 | |
5 | rncoss 4620 | . . 3 ⊢ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴 | |
6 | xpss12 4463 | . . 3 ⊢ ((dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵 ∧ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴) → (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ⊆ (dom 𝐵 × ran 𝐴)) | |
7 | 4, 5, 6 | mp2an 416 | . 2 ⊢ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ⊆ (dom 𝐵 × ran 𝐴) |
8 | 3, 7 | sstri 3008 | 1 ⊢ (𝐴 ∘ 𝐵) ⊆ (dom 𝐵 × ran 𝐴) |
Colors of variables: wff set class |
Syntax hints: ⊆ wss 2973 × cxp 4361 dom cdm 4363 ran crn 4364 ∘ ccom 4367 Rel wrel 4368 |
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 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 |
This theorem is referenced by: coexg 4882 tposssxp 5887 |
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