Metamath Proof Explorer

Theorem dff12

Description: Alternate definition of a one-to-one function. (Contributed by NM, 31-Dec-1996)

Ref Expression
Assertion dff12 
|- ( F : A -1-1-> B <-> ( F : A --> B /\ A. y E* x x F y ) )

Proof

Step Hyp Ref Expression
1 df-f1 
 |-  ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
2 funcnv2 
 |-  ( Fun `' F <-> A. y E* x x F y )
3 2 anbi2i 
 |-  ( ( F : A --> B /\ Fun `' F ) <-> ( F : A --> B /\ A. y E* x x F y ) )
4 1 3 bitri 
 |-  ( F : A -1-1-> B <-> ( F : A --> B /\ A. y E* x x F y ) )