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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 6106 | . . 3 ⊢ Rel (𝐴 ∘ 𝐵) | |
2 | relssdmrn 6266 | . . 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 5968 | . . 3 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵 | |
5 | rncoss 5969 | . . 3 ⊢ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴 | |
6 | xpss12 5687 | . . 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 691 | . 2 ⊢ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ⊆ (dom 𝐵 × ran 𝐴) |
8 | 3, 7 | sstri 3987 | 1 ⊢ (𝐴 ∘ 𝐵) ⊆ (dom 𝐵 × ran 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3944 × cxp 5670 dom cdm 5672 ran crn 5673 ∘ ccom 5676 Rel wrel 5677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 |
This theorem is referenced by: coexg 7931 tposssxp 8229 metustexhalf 24452 rtrclex 42970 trclexi 42973 rtrclexi 42974 cnvtrcl0 42979 |
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