Theorem isfsupp

Description: The property of a class to be a finitely supported function (in relation to a given zero). (Contributed by AV, 23-May-2019)

	Ref	Expression
Assertion	isfsupp	⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) → ( 𝑅 finSupp 𝑍 ↔ ( Fun 𝑅 ∧ ( 𝑅 supp 𝑍 ) ∈ Fin ) ) )

Proof

Step	Hyp	Ref	Expression
1		funeq	⊢ ( 𝑟 = 𝑅 → ( Fun 𝑟 ↔ Fun 𝑅 ) )
2	1	adantr	⊢ ( ( 𝑟 = 𝑅 ∧ 𝑧 = 𝑍 ) → ( Fun 𝑟 ↔ Fun 𝑅 ) )
3		oveq12	⊢ ( ( 𝑟 = 𝑅 ∧ 𝑧 = 𝑍 ) → ( 𝑟 supp 𝑧 ) = ( 𝑅 supp 𝑍 ) )
4	3	eleq1d	⊢ ( ( 𝑟 = 𝑅 ∧ 𝑧 = 𝑍 ) → ( ( 𝑟 supp 𝑧 ) ∈ Fin ↔ ( 𝑅 supp 𝑍 ) ∈ Fin ) )
5	2 4	anbi12d	⊢ ( ( 𝑟 = 𝑅 ∧ 𝑧 = 𝑍 ) → ( ( Fun 𝑟 ∧ ( 𝑟 supp 𝑧 ) ∈ Fin ) ↔ ( Fun 𝑅 ∧ ( 𝑅 supp 𝑍 ) ∈ Fin ) ) )
6		df-fsupp	⊢ finSupp = { ⟨ 𝑟 , 𝑧 ⟩ ∣ ( Fun 𝑟 ∧ ( 𝑟 supp 𝑧 ) ∈ Fin ) }
7	5 6	brabga	⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) → ( 𝑅 finSupp 𝑍 ↔ ( Fun 𝑅 ∧ ( 𝑅 supp 𝑍 ) ∈ Fin ) ) )