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Metamath Proof Explorer

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 )