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Mirrors > Home > MPE Home > Th. List > cossxp | Structured version Visualization version GIF version |
Description: Composition as a subset of the Cartesian 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 5875 | . . 3 ⊢ Rel (𝐴 ∘ 𝐵) | |
2 | relssdmrn 5898 | . . 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 5619 | . . 3 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵 | |
5 | rncoss 5620 | . . 3 ⊢ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴 | |
6 | xpss12 5358 | . . 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 685 | . 2 ⊢ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ⊆ (dom 𝐵 × ran 𝐴) |
8 | 3, 7 | sstri 3837 | 1 ⊢ (𝐴 ∘ 𝐵) ⊆ (dom 𝐵 × ran 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3799 × cxp 5341 dom cdm 5343 ran crn 5344 ∘ ccom 5347 Rel wrel 5348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4875 df-opab 4937 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 |
This theorem is referenced by: coexg 7380 tposssxp 7622 metustexhalf 22732 rtrclex 38766 trclexi 38769 rtrclexi 38770 cnvtrcl0 38775 |
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