Theorem f1ss 6106

Description: A function that is one-to-one is also one-to-one on some superset of its codomain. (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 6101	. . 3 |- ( F : A -1-1-> B -> F : A --> B )
2		fss 6056	. . 3 |- ( ( F : A --> B /\ B C_ C ) -> F : A --> C )
3	1, 2	sylan 488	. 2 |- ( ( F : A -1-1-> B /\ B C_ C ) -> F : A --> C )
4		df-f1 5893	. . . 4 |- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
5	4	simprbi 480	. . 3 |- ( F : A -1-1-> B -> Fun `' F )
6	5	adantr 481	. 2 |- ( ( F : A -1-1-> B /\ B C_ C ) -> Fun `' F )
7		df-f1 5893	. 2 |- ( F : A -1-1-> C <-> ( F : A --> C /\ Fun `' F ) )
8	3, 6, 7	sylanbrc 698	1 |- ( ( F : A -1-1-> B /\ B C_ C ) -> F : A -1-1-> C )

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

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-in 3581  df-ss 3588  df-f 5892  df-f1 5893

This theorem is referenced by:  f1sng  6178  f1prex  6539  domssex2  8120  1sdom  8163  marypha1lem  8339  marypha2  8345  isinffi  8818  fseqenlem1  8847  dfac12r  8968  ackbij2  9065  cff1  9080  fin23lem28  9162  fin23lem41  9174  pwfseqlem5  9485  hashf1lem1  13239  gsumzres  18310  gsumzcl2  18311  gsumzf1o  18313  gsumzaddlem  18321  gsumzmhm  18337  gsumzoppg  18344  lindfres  20162  islindf3  20165  dvne0f1  23775  istrkg2ld  25359  ausgrusgrb  26060  uspgrushgr  26070  usgruspgr  26073  uspgr1e  26136  sizusglecusglem1  26357  qqhre  30064  erdsze2lem1  31185  eldioph2lem2  37324  eldioph2  37325