Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 5037 | . . 3 ⊢ Rel (𝐴 ∘ 𝐵) | |
2 | relssdmrn 5059 | . . 3 ⊢ (Rel (𝐴 ∘ 𝐵) → (𝐴 ∘ 𝐵) ⊆ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (𝐴 ∘ 𝐵) ⊆ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) |
4 | dmcoss 4808 | . . 3 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵 | |
5 | rncoss 4809 | . . 3 ⊢ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴 | |
6 | xpss12 4646 | . . 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 422 | . 2 ⊢ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ⊆ (dom 𝐵 × ran 𝐴) |
8 | 3, 7 | sstri 3106 | 1 ⊢ (𝐴 ∘ 𝐵) ⊆ (dom 𝐵 × ran 𝐴) |
Colors of variables: wff set class |
Syntax hints: ⊆ wss 3071 × cxp 4537 dom cdm 4539 ran crn 4540 ∘ ccom 4543 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-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: cossxp2 5062 cocnvss 5064 coexg 5083 tposssxp 6146 |
Copyright terms: Public domain | W3C validator |