Theorem ffi

Description: A function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Assertion	ffi	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐴 ∈ Fin ) → 𝐹 ∈ Fin )

Proof

Step	Hyp	Ref	Expression
1		ffn	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → 𝐹 Fn 𝐴 )
2		fnfi	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin ) → 𝐹 ∈ Fin )
3	1 2	sylan	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐴 ∈ Fin ) → 𝐹 ∈ Fin )