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Metamath Proof Explorer

Theorem f1ocnv

Description: The converse of a one-to-one onto function is also one-to-one onto. (Contributed by NM, 11-Feb-1997) (Proof shortened by Andrew Salmon, 22-Oct-2011)

Ref Expression
Assertion f1ocnv 
|- ( F : A -1-1-onto-> B -> `' F : B -1-1-onto-> A )

Proof

Step Hyp Ref Expression
1 fnrel 
 |-  ( F Fn A -> Rel F )
2 dfrel2 
 |-  ( Rel F <-> `' `' F = F )
3 fneq1 
 |-  ( `' `' F = F -> ( `' `' F Fn A <-> F Fn A ) )
4 3 biimprd 
 |-  ( `' `' F = F -> ( F Fn A -> `' `' F Fn A ) )
5 2 4 sylbi 
 |-  ( Rel F -> ( F Fn A -> `' `' F Fn A ) )
6 1 5 mpcom 
 |-  ( F Fn A -> `' `' F Fn A )
7 6 anim1ci 
 |-  ( ( F Fn A /\ `' F Fn B ) -> ( `' F Fn B /\ `' `' F Fn A ) )
8 dff1o4 
 |-  ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) )
9 dff1o4 
 |-  ( `' F : B -1-1-onto-> A <-> ( `' F Fn B /\ `' `' F Fn A ) )
10 7 8 9 3imtr4i 
 |-  ( F : A -1-1-onto-> B -> `' F : B -1-1-onto-> A )