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 6536. (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 6536 | . 2 ⊢ (((𝐹 ↾ 𝐵):𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶) | |
4 | 1, 2, 3 | syl2anc 586 | 1 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐴⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↾ cres 5560 ∘ ccom 5562 ⟶wf 6354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-fun 6360 df-fn 6361 df-f 6362 |
This theorem is referenced by: extoimad 40521 imo72b2lem0 40522 imo72b2lem2 40524 imo72b2lem1 40527 imo72b2 40531 |
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