Theorem f1ss 5777

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 5772	. . 3 |- ( F : A -1-1-> B -> F : A --> B )
2		fss 5730	. . 3 |- ( ( F : A --> B /\ B C_ C ) -> F : A --> C )
3	1, 2	sylan 471	. 2 |- ( ( F : A -1-1-> B /\ B C_ C ) -> F : A --> C )
4		df-f1 5584	. . . 4 |- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
5	4	simprbi 464	. . 3 |- ( F : A -1-1-> B -> Fun `' F )
6	5	adantr 465	. 2 |- ( ( F : A -1-1-> B /\ B C_ C ) -> Fun `' F )
7		df-f1 5584	. 2 |- ( F : A -1-1-> C <-> ( F : A --> C /\ Fun `' F ) )
8	3, 6, 7	sylanbrc 664	1 |- ( ( F : A -1-1-> B /\ B C_ C ) -> F : A -1-1-> C )

Syntax hints:   -> wi 4   /\ wa 369   C_ wss 3469   `'ccnv 4991   Fun wfun 5573   -->wf 5575   -1-1->wf1 5576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-in 3476  df-ss 3483  df-f 5583  df-f1 5584

This theorem is referenced by:  domssex2  7667  1sdom  7712  marypha1lem  7882  marypha2  7888  isinffi  8362  fseqenlem1  8394  dfac12r  8515  ackbij2  8612  cff1  8627  fin23lem28  8709  fin23lem41  8721  pwfseqlem5  9030  hashf1lem1  12457  gsumzres  16698  gsumzcl2  16699  gsumzf1o  16701  gsumzresOLD  16702  gsumzclOLD  16703  gsumzf1oOLD  16704  gsumzaddlem  16718  gsumzaddlemOLD  16720  gsumzmhm  16741  gsumzmhmOLD  16742  gsumzoppg  16751  gsumzoppgOLD  16752  lindfres  18618  islindf3  18621  dvne0f1  22141  istrkg2ld  23579  ausisusgra  24018  usisuslgra  24027  uslgra1  24034  usgra1  24035  sizeusglecusglem1  24146  2trllemE  24217  constr1trl  24252  qqhre  27484  erdsze2lem1  28137  eldioph2lem2  30149  eldioph2  30150  frgrancvvdeqlem8  31759