Theorem f1ss 5257

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 5251	. . 3 |- ( F : A -1-1-> B -> F : A --> B )
2		fss 5207	. . 3 |- ( ( F : A --> B /\ B C_ C ) -> F : A --> C )
3	1, 2	sylan 278	. 2 |- ( ( F : A -1-1-> B /\ B C_ C ) -> F : A --> C )
4		df-f1 5054	. . . 4 |- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
5	4	simprbi 270	. . 3 |- ( F : A -1-1-> B -> Fun `' F )
6	5	adantr 271	. 2 |- ( ( F : A -1-1-> B /\ B C_ C ) -> Fun `' F )
7		df-f1 5054	. 2 |- ( F : A -1-1-> C <-> ( F : A --> C /\ Fun `' F ) )
8	3, 6, 7	sylanbrc 409	1 |- ( ( F : A -1-1-> B /\ B C_ C ) -> F : A -1-1-> C )

Syntax hints:   -> wi 4   /\ wa 103   C_ wss 3013   `'ccnv 4466   Fun wfun 5043   -->wf 5045   -1-1->wf1 5046

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-11 1449  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-in 3019  df-ss 3026  df-f 5053  df-f1 5054

This theorem is referenced by: (None)