Theorem ffvelrnd

Description: A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016)

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

Proof

Step	Hyp	Ref	Expression
1		ffvelrnd.1	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
2		ffvelrnd.2	⊢ ( 𝜑 → 𝐶 ∈ 𝐴 )
3	1	ffvelrnda	⊢ ( ( 𝜑 ∧ 𝐶 ∈ 𝐴 ) → ( 𝐹 ‘ 𝐶 ) ∈ 𝐵 )
4	2 3	mpdan	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐶 ) ∈ 𝐵 )