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 ) ) )