Theorem f1ssr 5118

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 5113	. . . 4 |- ( F : A -1-1-> B -> F Fn A )
2	1	adantr 270	. . 3 |- ( ( F : A -1-1-> B /\ ran F C_ C ) -> F Fn A )
3		simpr 108	. . 3 |- ( ( F : A -1-1-> B /\ ran F C_ C ) -> ran F C_ C )
4		df-f 4926	. . 3 |- ( F : A --> C <-> ( F Fn A /\ ran F C_ C ) )
5	2, 3, 4	sylanbrc 408	. 2 |- ( ( F : A -1-1-> B /\ ran F C_ C ) -> F : A --> C )
6		df-f1 4927	. . . 4 |- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
7	6	simprbi 269	. . 3 |- ( F : A -1-1-> B -> Fun `' F )
8	7	adantr 270	. 2 |- ( ( F : A -1-1-> B /\ ran F C_ C ) -> Fun `' F )
9		df-f1 4927	. 2 |- ( F : A -1-1-> C <-> ( F : A --> C /\ Fun `' F ) )
10	5, 8, 9	sylanbrc 408	1 |- ( ( F : A -1-1-> B /\ ran F C_ C ) -> F : A -1-1-> C )

Syntax hints:   -> wi 4   /\ wa 102   C_ wss 2973   `'ccnv 4362   ran crn 4364   Fun wfun 4916   Fn wfn 4917   -->wf 4918   -1-1->wf1 4919

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115  df-f 4926  df-f1 4927

This theorem is referenced by: (None)