Theorem f1ss 5117

Description: A function that is one-to-one is also one-to-one on some superset of its range. (Contributed by Mario Carneiro, 12-Jan-2013.)

Assertion

Ref	Expression
f1ss	|- ( ( F : A -1-1-> B /\ B C_ C ) -> F : A -1-1-> C )

Proof of Theorem f1ss

Step	Hyp	Ref	Expression
1		f1f 5112	. . 3 |- ( F : A -1-1-> B -> F : A --> B )
2		fss 5074	. . 3 |- ( ( F : A --> B /\ B C_ C ) -> F : A --> C )
3	1, 2	sylan 277	. 2 |- ( ( F : A -1-1-> B /\ B C_ C ) -> F : A --> C )
4		df-f1 4927	. . . 4 |- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
5	4	simprbi 269	. . 3 |- ( F : A -1-1-> B -> Fun `' F )
6	5	adantr 270	. 2 |- ( ( F : A -1-1-> B /\ B C_ C ) -> Fun `' F )
7		df-f1 4927	. 2 |- ( F : A -1-1-> C <-> ( F : A --> C /\ Fun `' F ) )
8	3, 6, 7	sylanbrc 408	1 |- ( ( F : A -1-1-> B /\ B C_ C ) -> F : A -1-1-> C )

Syntax hints:   -> wi 4   /\ wa 102   C_ wss 2973   `'ccnv 4362   Fun wfun 4916   -->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  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-in 2979  df-ss 2986  df-f 4926  df-f1 4927

This theorem is referenced by: (None)