Theorem dff1o5

Description: Alternate definition of one-to-one onto function. (Contributed by NM, 10-Dec-2003) (Proof shortened by Andrew Salmon, 22-Oct-2011)

	Ref	Expression
Assertion	dff1o5	|- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ ran F = B ) )

Proof

Step	Hyp	Ref	Expression
1		df-f1o	|- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
2		dffo2	|- ( F : A -onto-> B <-> ( F : A --> B /\ ran F = B ) )
3		f1f	|- ( F : A -1-1-> B -> F : A --> B )
4	3	biantrurd	|- ( F : A -1-1-> B -> ( ran F = B <-> ( F : A --> B /\ ran F = B ) ) )
5	2 4	bitr4id	|- ( F : A -1-1-> B -> ( F : A -onto-> B <-> ran F = B ) )
6	5	pm5.32i	|- ( ( F : A -1-1-> B /\ F : A -onto-> B ) <-> ( F : A -1-1-> B /\ ran F = B ) )
7	1 6	bitri	|- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ ran F = B ) )