Theorem f1ssr 5400

Description: Combine a one-to-one function with a restriction on the domain. (Contributed by Stefan O'Rear, 20-Feb-2015.)

Assertion

Ref	Expression
f1ssr	|- ( ( F : A -1-1-> B /\ ran F C_ C ) -> F : A -1-1-> C )

Proof of Theorem f1ssr

Step	Hyp	Ref	Expression
1		f1fn 5395	. . . 4 |- ( F : A -1-1-> B -> F Fn A )
2	1	adantr 274	. . 3 |- ( ( F : A -1-1-> B /\ ran F C_ C ) -> F Fn A )
3		simpr 109	. . 3 |- ( ( F : A -1-1-> B /\ ran F C_ C ) -> ran F C_ C )
4		df-f 5192	. . 3 |- ( F : A --> C <-> ( F Fn A /\ ran F C_ C ) )
5	2, 3, 4	sylanbrc 414	. 2 |- ( ( F : A -1-1-> B /\ ran F C_ C ) -> F : A --> C )
6		df-f1 5193	. . . 4 |- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
7	6	simprbi 273	. . 3 |- ( F : A -1-1-> B -> Fun `' F )
8	7	adantr 274	. 2 |- ( ( F : A -1-1-> B /\ ran F C_ C ) -> Fun `' F )
9		df-f1 5193	. 2 |- ( F : A -1-1-> C <-> ( F : A --> C /\ Fun `' F ) )
10	5, 8, 9	sylanbrc 414	1 |- ( ( F : A -1-1-> B /\ ran F C_ C ) -> F : A -1-1-> C )

Syntax hints:   -> wi 4   /\ wa 103   C_ wss 3116   `'ccnv 4603   ran crn 4605   Fun wfun 5182   Fn wfn 5183   -->wf 5184   -1-1->wf1 5185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-f 5192  df-f1 5193

This theorem is referenced by:  f1ff1  5401  difinfsn  7065