Theorem dffn4

Description: A function maps onto its range. (Contributed by NM, 10-May-1998)

	Ref	Expression
Assertion	dffn4	⊢ ( 𝐹 Fn 𝐴 ↔ 𝐹 : 𝐴 –onto→ ran 𝐹 )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ran 𝐹 = ran 𝐹
2	1	biantru	⊢ ( 𝐹 Fn 𝐴 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = ran 𝐹 ) )
3		df-fo	⊢ ( 𝐹 : 𝐴 –onto→ ran 𝐹 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = ran 𝐹 ) )
4	2 3	bitr4i	⊢ ( 𝐹 Fn 𝐴 ↔ 𝐹 : 𝐴 –onto→ ran 𝐹 )