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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 6097 | . . 3 ⊢ Rel (𝐴 ∘ 𝐵) | |
2 | relssdmrn 6121 | . . 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 5842 | . . 3 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵 | |
5 | rncoss 5843 | . . 3 ⊢ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴 | |
6 | xpss12 5570 | . . 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 690 | . 2 ⊢ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ⊆ (dom 𝐵 × ran 𝐴) |
8 | 3, 7 | sstri 3976 | 1 ⊢ (𝐴 ∘ 𝐵) ⊆ (dom 𝐵 × ran 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3936 × cxp 5553 dom cdm 5555 ran crn 5556 ∘ ccom 5559 Rel wrel 5560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 |
This theorem is referenced by: coexg 7634 tposssxp 7896 metustexhalf 23166 rtrclex 39997 trclexi 40000 rtrclexi 40001 cnvtrcl0 40006 |
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