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Metamath Proof Explorer

Theorem fvco3

Description: Value of a function composition. (Contributed by NM, 3-Jan-2004) (Revised by Mario Carneiro, 26-Dec-2014)

		Ref	Expression
	Assertion	fvco3	⊢ ( ( 𝐺 : 𝐴 ⟶ 𝐵 ∧ 𝐶 ∈ 𝐴 ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝐶 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝐶 ) ) )

Step	Hyp	Ref	Expression
1		ffn	⊢ ( 𝐺 : 𝐴 ⟶ 𝐵 → 𝐺 Fn 𝐴 )
2		fvco2	⊢ ( ( 𝐺 Fn 𝐴 ∧ 𝐶 ∈ 𝐴 ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝐶 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝐶 ) ) )
3	1 2	sylan	⊢ ( ( 𝐺 : 𝐴 ⟶ 𝐵 ∧ 𝐶 ∈ 𝐴 ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝐶 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝐶 ) ) )