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 6059. (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 6059 | . . 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 715 | 1 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐴⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ↾ cres 5116 ∘ 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-res 5126 df-fun 5890 df-fn 5891 df-f 5892 |
This theorem is referenced by: extoimad 38464 imo72b2lem0 38465 imo72b2lem2 38467 imo72b2lem1 38471 imo72b2 38475 |
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