Theorem fisuppfi

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

	Ref	Expression
Hypotheses	fisuppfi.1	$$⊢\phi \to A\in \mathrm{Fin}$$
	fisuppfi.2	$$⊢\phi \to F:A⟶B$$
Assertion	fisuppfi	$$⊢\phi \to F^{-1}\left[C\right]\in \mathrm{Fin}$$

Proof

Step	Hyp	Ref	Expression
1		fisuppfi.1	$$⊢\phi \to A\in \mathrm{Fin}$$
2		fisuppfi.2	$$⊢\phi \to F:A⟶B$$
3		cnvimass	$$⊢F^{-1}\left[C\right]\subseteq \mathrm{dom}F$$
4	3 2	fssdm	$$⊢\phi \to F^{-1}\left[C\right]\subseteq A$$
5	1 4	ssfid	$$⊢\phi \to F^{-1}\left[C\right]\in \mathrm{Fin}$$