Theorem f1ssr 5098

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 5093	. . . 4 |- ( F : A -1-1-> B -> F Fn A )
2	1	adantr 261	. . 3 |- ( ( F : A -1-1-> B /\ ran F C_ C ) -> F Fn A )
3		simpr 103	. . 3 |- ( ( F : A -1-1-> B /\ ran F C_ C ) -> ran F C_ C )
4		df-f 4906	. . 3 |- ( F : A --> C <-> ( F Fn A /\ ran F C_ C ) )
5	2, 3, 4	sylanbrc 394	. 2 |- ( ( F : A -1-1-> B /\ ran F C_ C ) -> F : A --> C )
6		df-f1 4907	. . . 4 |- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
7	6	simprbi 260	. . 3 |- ( F : A -1-1-> B -> Fun `' F )
8	7	adantr 261	. 2 |- ( ( F : A -1-1-> B /\ ran F C_ C ) -> Fun `' F )
9		df-f1 4907	. 2 |- ( F : A -1-1-> C <-> ( F : A --> C /\ Fun `' F ) )
10	5, 8, 9	sylanbrc 394	1 |- ( ( F : A -1-1-> B /\ ran F C_ C ) -> F : A -1-1-> C )

Syntax hints:   -> wi 4   /\ wa 97   C_ wss 2917   `'ccnv 4344   ran crn 4346   Fun wfun 4896   Fn wfn 4897   -->wf 4898   -1-1->wf1 4899

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110  df-f 4906  df-f1 4907

This theorem is referenced by: (None)