Theorem dff1o4

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	dff1o4	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ ◡ 𝐹 Fn 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		dff1o2	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ Fun ◡ 𝐹 ∧ ran 𝐹 = 𝐵 ) )
2		3anass	⊢ ( ( 𝐹 Fn 𝐴 ∧ Fun ◡ 𝐹 ∧ ran 𝐹 = 𝐵 ) ↔ ( 𝐹 Fn 𝐴 ∧ ( Fun ◡ 𝐹 ∧ ran 𝐹 = 𝐵 ) ) )
3		df-rn	⊢ ran 𝐹 = dom ◡ 𝐹
4	3	eqeq1i	⊢ ( ran 𝐹 = 𝐵 ↔ dom ◡ 𝐹 = 𝐵 )
5	4	anbi2i	⊢ ( ( Fun ◡ 𝐹 ∧ ran 𝐹 = 𝐵 ) ↔ ( Fun ◡ 𝐹 ∧ dom ◡ 𝐹 = 𝐵 ) )
6		df-fn	⊢ ( ◡ 𝐹 Fn 𝐵 ↔ ( Fun ◡ 𝐹 ∧ dom ◡ 𝐹 = 𝐵 ) )
7	5 6	bitr4i	⊢ ( ( Fun ◡ 𝐹 ∧ ran 𝐹 = 𝐵 ) ↔ ◡ 𝐹 Fn 𝐵 )
8	7	anbi2i	⊢ ( ( 𝐹 Fn 𝐴 ∧ ( Fun ◡ 𝐹 ∧ ran 𝐹 = 𝐵 ) ) ↔ ( 𝐹 Fn 𝐴 ∧ ◡ 𝐹 Fn 𝐵 ) )
9	1 2 8	3bitri	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ ◡ 𝐹 Fn 𝐵 ) )