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Metamath Proof Explorer

Theorem f1imapss

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

Ref Expression
Assertion f1imapss 
|- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " C ) C. ( F " D ) <-> C 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 f1imaeq 
 |-  ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " C ) = ( F " D ) <-> C = D ) )
3 2 notbid 
 |-  ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( -. ( F " C ) = ( F " D ) <-> -. C = D ) )
4 1 3 anbi12d 
 |-  ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( ( F " C ) C_ ( F " D ) /\ -. ( F " C ) = ( F " D ) ) <-> ( C C_ D /\ -. C = D ) ) )
5 dfpss2 
 |-  ( ( F " C ) C. ( F " D ) <-> ( ( F " C ) C_ ( F " D ) /\ -. ( F " C ) = ( F " D ) ) )
6 dfpss2 
 |-  ( C C. D <-> ( C C_ D /\ -. C = D ) )
7 4 5 6 3bitr4g 
 |-  ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " C ) C. ( F " D ) <-> C C. D ) )