Theorem fsuppimp

Description: Implications of a class being a finitely supported function (in relation to a given zero). (Contributed by AV, 26-May-2019)

	Ref	Expression
Assertion	fsuppimp	⊢ ( 𝑅 finSupp 𝑍 → ( Fun 𝑅 ∧ ( 𝑅 supp 𝑍 ) ∈ Fin ) )

Proof

Step	Hyp	Ref	Expression
1		relfsupp	⊢ Rel finSupp
2	1	brrelex12i	⊢ ( 𝑅 finSupp 𝑍 → ( 𝑅 ∈ V ∧ 𝑍 ∈ V ) )
3		isfsupp	⊢ ( ( 𝑅 ∈ V ∧ 𝑍 ∈ V ) → ( 𝑅 finSupp 𝑍 ↔ ( Fun 𝑅 ∧ ( 𝑅 supp 𝑍 ) ∈ Fin ) ) )
4	3	biimpd	⊢ ( ( 𝑅 ∈ V ∧ 𝑍 ∈ V ) → ( 𝑅 finSupp 𝑍 → ( Fun 𝑅 ∧ ( 𝑅 supp 𝑍 ) ∈ Fin ) ) )
5	2 4	mpcom	⊢ ( 𝑅 finSupp 𝑍 → ( Fun 𝑅 ∧ ( 𝑅 supp 𝑍 ) ∈ Fin ) )