Theorem f1ssr 6107

Description: A function that is one-to-one is also one-to-one on some superset of its range. (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 6102	. . . 4 |- ( F : A -1-1-> B -> F Fn A )
2	1	adantr 481	. . 3 |- ( ( F : A -1-1-> B /\ ran F C_ C ) -> F Fn A )
3		simpr 477	. . 3 |- ( ( F : A -1-1-> B /\ ran F C_ C ) -> ran F C_ C )
4		df-f 5892	. . 3 |- ( F : A --> C <-> ( F Fn A /\ ran F C_ C ) )
5	2, 3, 4	sylanbrc 698	. 2 |- ( ( F : A -1-1-> B /\ ran F C_ C ) -> F : A --> C )
6		df-f1 5893	. . . 4 |- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
7	6	simprbi 480	. . 3 |- ( F : A -1-1-> B -> Fun `' F )
8	7	adantr 481	. 2 |- ( ( F : A -1-1-> B /\ ran F C_ C ) -> Fun `' F )
9		df-f1 5893	. 2 |- ( F : A -1-1-> C <-> ( F : A --> C /\ Fun `' F ) )
10	5, 8, 9	sylanbrc 698	1 |- ( ( F : A -1-1-> B /\ ran F C_ C ) -> F : A -1-1-> C )

Syntax hints:   -> wi 4   /\ wa 384   C_ wss 3574   `'ccnv 5113   ran crn 5115   Fun wfun 5882   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-an 386  df-f 5892  df-f1 5893

This theorem is referenced by:  domdifsn  8043  marypha1  8340  m2cpmf1  20548  ausgrusgri  26063  uspgrupgrushgr  26072  usgrumgruspgr  26075  usgruspgrb  26076  usgrres  26200  usgrres1  26207