Theorem efmndfv

Description: The function value of an endofunction. (Contributed by AV, 27-Jan-2024)

	Ref	Expression
Hypotheses	efmndbas.g	⊢ 𝐺 = ( EndoFMnd ‘ 𝐴 )
	efmndbas.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
Assertion	efmndfv	⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑋 ) ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		efmndbas.g	⊢ 𝐺 = ( EndoFMnd ‘ 𝐴 )
2		efmndbas.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
3	1 2	efmndbasf	⊢ ( 𝐹 ∈ 𝐵 → 𝐹 : 𝐴 ⟶ 𝐴 )
4	3	ffvelcdmda	⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑋 ) ∈ 𝐴 )