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Mirrors > Home > MPE Home > Th. List > Mathboxes > fcoss | Structured version Visualization version GIF version |
Description: Composition of two mappings. Similar to fco 6526, but with a weaker condition on the domain of 𝐹. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
fcoss.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
fcoss.c | ⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
fcoss.g | ⊢ (𝜑 → 𝐺:𝐷⟶𝐶) |
Ref | Expression |
---|---|
fcoss | ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐷⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fcoss.f | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
2 | fcoss.g | . . 3 ⊢ (𝜑 → 𝐺:𝐷⟶𝐶) | |
3 | fcoss.c | . . 3 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) | |
4 | 2, 3 | fssd 6523 | . 2 ⊢ (𝜑 → 𝐺:𝐷⟶𝐴) |
5 | fco 6526 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐷⟶𝐴) → (𝐹 ∘ 𝐺):𝐷⟶𝐵) | |
6 | 1, 4, 5 | syl2anc 586 | 1 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐷⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3936 ∘ ccom 5554 ⟶wf 6346 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-fun 6352 df-fn 6353 df-f 6354 |
This theorem is referenced by: volicoff 42273 voliooicof 42274 ovolval2 42919 ovolval5lem2 42928 ovnovollem1 42931 ovnovollem2 42932 |
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