Theorem dffn2

Description: Any function is a mapping into _V . (Contributed by NM, 31-Oct-1995) (Proof shortened by Andrew Salmon, 17-Sep-2011)

	Ref	Expression
Assertion	dffn2	⊢ ( 𝐹 Fn 𝐴 ↔ 𝐹 : 𝐴 ⟶ V )

Proof

Step	Hyp	Ref	Expression
1		ssv	⊢ ran 𝐹 ⊆ V
2	1	biantru	⊢ ( 𝐹 Fn 𝐴 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ V ) )
3		df-f	⊢ ( 𝐹 : 𝐴 ⟶ V ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ V ) )
4	2 3	bitr4i	⊢ ( 𝐹 Fn 𝐴 ↔ 𝐹 : 𝐴 ⟶ V )