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Metamath Proof Explorer

Theorem fvco3d

Description: Value of a function composition. Deduction form of fvco3 . (Contributed by Stanislas Polu, 9-Mar-2020)

Ref Expression
Hypotheses fvco3d.1 ⊢ ( 𝜑 → 𝐺 : 𝐴 ⟶ 𝐵 )
fvco3d.2 ⊢ ( 𝜑 → 𝐶 ∈ 𝐴 )
Assertion fvco3d ( 𝜑 → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝐶 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 fvco3d.1 ⊢ ( 𝜑 → 𝐺 : 𝐴 ⟶ 𝐵 )
2 fvco3d.2 ⊢ ( 𝜑 → 𝐶 ∈ 𝐴 )
3 fvco3 ⊢ ( ( 𝐺 : 𝐴 ⟶ 𝐵 ∧ 𝐶 ∈ 𝐴 ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝐶 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝐶 ) ) )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝐶 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝐶 ) ) )