Metamath Proof Explorer

Theorem f1imaeq

Description: Taking images under a one-to-one function preserves equality. (Contributed by Stefan O'Rear, 30-Oct-2014)

Ref Expression
Assertion f1imaeq 
|- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " C ) = ( F " D ) <-> C = D ) )

Proof

Step Hyp Ref Expression
1 f1imass 
 |-  ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " C ) C_ ( F " D ) <-> C C_ D ) )
2 f1imass 
 |-  ( ( F : A -1-1-> B /\ ( D C_ A /\ C C_ A ) ) -> ( ( F " D ) C_ ( F " C ) <-> D C_ C ) )
3 2 ancom2s 
 |-  ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " D ) C_ ( F " C ) <-> D C_ C ) )
4 1 3 anbi12d 
 |-  ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( ( F " C ) C_ ( F " D ) /\ ( F " D ) C_ ( F " C ) ) <-> ( C C_ D /\ D C_ C ) ) )
5 eqss 
 |-  ( ( F " C ) = ( F " D ) <-> ( ( F " C ) C_ ( F " D ) /\ ( F " D ) C_ ( F " C ) ) )
6 eqss 
 |-  ( C = D <-> ( C C_ D /\ D C_ C ) )
7 4 5 6 3bitr4g 
 |-  ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " C ) = ( F " D ) <-> C = D ) )