Theorem wfximgfd

Description: The value of a function on its domain is in the image of the function. (Contributed by Stanislas Polu, 9-Mar-2020)

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

Proof

Step	Hyp	Ref	Expression
1		wfximgfd.1	⊢ ( 𝜑 → 𝐶 ∈ 𝐴 )
2		wfximgfd.2	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
3	2	ffnd	⊢ ( 𝜑 → 𝐹 Fn 𝐴 )
4	3 1 1	fnfvimad	⊢ ( 𝜑 → ( 𝐹 ‘ 𝐶 ) ∈ ( 𝐹 “ 𝐴 ) )