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Metamath Proof Explorer

Theorem f1ssres

Description: A function that is one-to-one is also one-to-one on any subclass of its domain. (Contributed by Mario Carneiro, 17-Jan-2015)

Ref Expression
Assertion f1ssres 
|- ( ( F : A -1-1-> B /\ C C_ A ) -> ( F |` C ) : C -1-1-> B )

Proof

Step Hyp Ref Expression
1 f1f 
 |-  ( F : A -1-1-> B -> F : A --> B )
2 fssres 
 |-  ( ( F : A --> B /\ C C_ A ) -> ( F |` C ) : C --> B )
3 1 2 sylan 
 |-  ( ( F : A -1-1-> B /\ C C_ A ) -> ( F |` C ) : C --> B )
4 df-f1 
 |-  ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
5 funres11 
 |-  ( Fun `' F -> Fun `' ( F |` C ) )
6 4 5 simplbiim 
 |-  ( F : A -1-1-> B -> Fun `' ( F |` C ) )
7 6 adantr 
 |-  ( ( F : A -1-1-> B /\ C C_ A ) -> Fun `' ( F |` C ) )
8 df-f1 
 |-  ( ( F |` C ) : C -1-1-> B <-> ( ( F |` C ) : C --> B /\ Fun `' ( F |` C ) ) )
9 3 7 8 sylanbrc 
 |-  ( ( F : A -1-1-> B /\ C C_ A ) -> ( F |` C ) : C -1-1-> B )