Theorem dff1o3

Description: Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998) (Proof shortened by Andrew Salmon, 22-Oct-2011)

	Ref	Expression
Assertion	dff1o3	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ↔ ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ Fun ◡ 𝐹 ) )

Proof

Step	Hyp	Ref	Expression
1		3anan32	⊢ ( ( 𝐹 Fn 𝐴 ∧ Fun ◡ 𝐹 ∧ ran 𝐹 = 𝐵 ) ↔ ( ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ) ∧ Fun ◡ 𝐹 ) )
2		dff1o2	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ Fun ◡ 𝐹 ∧ ran 𝐹 = 𝐵 ) )
3		df-fo	⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ) )
4	3	anbi1i	⊢ ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ Fun ◡ 𝐹 ) ↔ ( ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ) ∧ Fun ◡ 𝐹 ) )
5	1 2 4	3bitr4i	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ↔ ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ Fun ◡ 𝐹 ) )