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Mathbox for Glauco Siliprandi |
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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 6747, 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 6740 | . 2 ⊢ (𝜑 → 𝐺:𝐷⟶𝐴) |
5 | fco 6747 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐷⟶𝐴) → (𝐹 ∘ 𝐺):𝐷⟶𝐵) | |
6 | 1, 4, 5 | syl2anc 583 | 1 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐷⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3947 ∘ ccom 5682 ⟶wf 6544 |
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-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
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-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-fun 6550 df-fn 6551 df-f 6552 |
This theorem is referenced by: volicoff 45383 voliooicof 45384 ovolval2 46032 ovolval5lem2 46041 ovolval5lem3 46042 ovnovollem1 46044 ovnovollem2 46045 |
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