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Mathbox for Stanislas Polu |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fco2d | Structured version Visualization version GIF version |
Description: Natural deduction form of fco2 6309. (Contributed by Stanislas Polu, 9-Mar-2020.) |
Ref | Expression |
---|---|
fco2d.1 | ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) |
fco2d.2 | ⊢ (𝜑 → (𝐹 ↾ 𝐵):𝐵⟶𝐶) |
Ref | Expression |
---|---|
fco2d | ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐴⟶𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fco2d.2 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝐵):𝐵⟶𝐶) | |
2 | fco2d.1 | . 2 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) | |
3 | fco2 6309 | . . 3 ⊢ (((𝐹 ↾ 𝐵):𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶) | |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → (((𝐹 ↾ 𝐵):𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶)) |
5 | 1, 2, 4 | mp2and 689 | 1 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐴⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ↾ cres 5357 ∘ ccom 5359 ⟶wf 6131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-br 4887 df-opab 4949 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-fun 6137 df-fn 6138 df-f 6139 |
This theorem is referenced by: extoimad 39413 imo72b2lem0 39414 imo72b2lem2 39416 imo72b2lem1 39420 imo72b2 39424 |
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