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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 6058, 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 6057 | . 2 ⊢ (𝜑 → 𝐺:𝐷⟶𝐴) |
5 | fco 6058 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐷⟶𝐴) → (𝐹 ∘ 𝐺):𝐷⟶𝐵) | |
6 | 1, 4, 5 | syl2anc 693 | 1 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐷⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3574 ∘ ccom 5118 ⟶wf 5884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-fun 5890 df-fn 5891 df-f 5892 |
This theorem is referenced by: volicoff 40212 voliooicof 40213 ovolval2 40858 ovolval5lem2 40867 ovnovollem1 40870 ovnovollem2 40871 |
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