Metamath Proof Explorer

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 ) )