Theorem dffn3

Description: A function maps to its range. (Contributed by NM, 1-Sep-1999)

	Ref	Expression
Assertion	dffn3	⊢ ( 𝐹 Fn 𝐴 ↔ 𝐹 : 𝐴 ⟶ ran 𝐹 )

Proof

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