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 )