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Metamath Proof Explorer

Theorem fcod

Description: Composition of two mappings. (Contributed by Glauco Siliprandi, 26-Jun-2021)

Ref Expression
Hypotheses fcod.1 ⊢ ( 𝜑 → 𝐹 : 𝐵 ⟶ 𝐶 )
fcod.2 ⊢ ( 𝜑 → 𝐺 : 𝐴 ⟶ 𝐵 )
Assertion fcod ( 𝜑 → ( 𝐹 ∘ 𝐺 ) : 𝐴 ⟶ 𝐶 )

Proof

Step Hyp Ref Expression
1 fcod.1 ⊢ ( 𝜑 → 𝐹 : 𝐵 ⟶ 𝐶 )
2 fcod.2 ⊢ ( 𝜑 → 𝐺 : 𝐴 ⟶ 𝐵 )
3 fco ⊢ ( ( 𝐹 : 𝐵 ⟶ 𝐶 ∧ 𝐺 : 𝐴 ⟶ 𝐵 ) → ( 𝐹 ∘ 𝐺 ) : 𝐴 ⟶ 𝐶 )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ( 𝐹 ∘ 𝐺 ) : 𝐴 ⟶ 𝐶 )