Theorem fisuppfi

Description: A function on a finite set is finitely supported. (Contributed by Mario Carneiro, 20-Jun-2015)

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

Proof

Step	Hyp	Ref	Expression
1		fisuppfi.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
2		fisuppfi.2	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
3		cnvimass	⊢ ( ◡ 𝐹 “ 𝐶 ) ⊆ dom 𝐹
4	3 2	fssdm	⊢ ( 𝜑 → ( ◡ 𝐹 “ 𝐶 ) ⊆ 𝐴 )
5	1 4	ssfid	⊢ ( 𝜑 → ( ◡ 𝐹 “ 𝐶 ) ∈ Fin )