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Metamath Proof Explorer

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 ) )