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Mirrors > Home > MPE Home > Th. List > fcod | Structured version Visualization version GIF version |
Description: Composition of two mappings. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
fcod.1 | ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) |
fcod.2 | ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) |
Ref | Expression |
---|---|
fcod | ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐴⟶𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fcod.1 | . 2 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) | |
2 | fcod.2 | . 2 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) | |
3 | fco 6531 | . 2 ⊢ ((𝐹: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 ∘ ccom 5559 ⟶wf 6351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-fun 6357 df-fn 6358 df-f 6359 |
This theorem is referenced by: selvval2lem4 39156 |
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