Theorem f1ss 5409

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 5403	. . 3 |- ( F : A -1-1-> B -> F : A --> B )
2		fss 5359	. . 3 |- ( ( F : A --> B /\ B C_ C ) -> F : A --> C )
3	1, 2	sylan 281	. 2 |- ( ( F : A -1-1-> B /\ B C_ C ) -> F : A --> C )
4		df-f1 5203	. . . 4 |- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
5	4	simprbi 273	. . 3 |- ( F : A -1-1-> B -> Fun `' F )
6	5	adantr 274	. 2 |- ( ( F : A -1-1-> B /\ B C_ C ) -> Fun `' F )
7		df-f1 5203	. 2 |- ( F : A -1-1-> C <-> ( F : A --> C /\ Fun `' F ) )
8	3, 6, 7	sylanbrc 415	1 |- ( ( F : A -1-1-> B /\ B C_ C ) -> F : A -1-1-> C )

Syntax hints:   -> wi 4   /\ wa 103   C_ wss 3121   `'ccnv 4610   Fun wfun 5192   -->wf 5194   -1-1->wf1 5195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-in 3127  df-ss 3134  df-f 5202  df-f1 5203

This theorem is referenced by:  f1sng  5484