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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 6763. (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 6763 | . 2 ⊢ (((𝐹 ↾ 𝐵):𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶) | |
4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐴⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↾ cres 5691 ∘ ccom 5693 ⟶wf 6559 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-fun 6565 df-fn 6566 df-f 6567 |
This theorem is referenced by: extoimad 44154 imo72b2lem0 44155 imo72b2lem2 44157 imo72b2lem1 44159 imo72b2 44162 |
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