Theorem f1ss 5469

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 5463	. . 3 |- ( F : A -1-1-> B -> F : A --> B )
2		fss 5419	. . 3 |- ( ( F : A --> B /\ B C_ C ) -> F : A --> C )
3	1, 2	sylan 283	. 2 |- ( ( F : A -1-1-> B /\ B C_ C ) -> F : A --> C )
4		df-f1 5263	. . . 4 |- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
5	4	simprbi 275	. . 3 |- ( F : A -1-1-> B -> Fun `' F )
6	5	adantr 276	. 2 |- ( ( F : A -1-1-> B /\ B C_ C ) -> Fun `' F )
7		df-f1 5263	. 2 |- ( F : A -1-1-> C <-> ( F : A --> C /\ Fun `' F ) )
8	3, 6, 7	sylanbrc 417	1 |- ( ( F : A -1-1-> B /\ B C_ C ) -> F : A -1-1-> C )

Syntax hints:   -> wi 4   /\ wa 104   C_ wss 3157   `'ccnv 4662   Fun wfun 5252   -->wf 5254   -1-1->wf1 5255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-in 3163  df-ss 3170  df-f 5262  df-f1 5263

This theorem is referenced by:  f1sng  5546