Theorem f1of1 5582

Description: A one-to-one onto mapping is a one-to-one mapping. (Contributed by NM, 12-Dec-2003.)

Assertion

Ref	Expression
f1of1	|- ( F : A -1-1-onto-> B -> F : A -1-1-> B )

Proof of Theorem f1of1

Step	Hyp	Ref	Expression
1		df-f1o 5333	. 2 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
2	1	simplbi 274	1 |- ( F : A -1-1-onto-> B -> F : A -1-1-> B )

Syntax hints:   -> wi 4   -1-1->wf1 5323   -onto->wfo 5324   -1-1-onto->wf1o 5325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106

This theorem depends on definitions:  df-bi 117  df-f1o 5333

This theorem is referenced by:  f1of  5583  f1sng  5627  f1oresrab  5812  f1ocnvfvrneq  5923  isores3  5956  isoini2  5960  f1oiso  5967  f1opw2  6229  tposf12  6435  enssdom  6935  mapen  7032  ssenen  7037  phplem4  7041  phplem4on  7054  fidceq  7056  en2eqpr  7099  fiintim  7123  f1finf1o  7146  preimaf1ofi  7150  isotilem  7205  inresflem  7259  casefun  7284  endjusym  7295  pr2cv1  7400  dju1p1e2  7408  frec2uzled  10691  iseqf1olemnab  10763  iseqf1olemab  10764  iseqf1olemnanb  10765  seqf1oglem1  10781  hashen  11046  hashfacen  11100  negfi  11789  fisumss  11954  fprodssdc  12152  phimullem  12798  eulerthlemh  12804  ctinfom  13050  ssnnctlemct  13068  f1ocpbllem  13394  f1ovscpbl  13396  xpsff1o2  13435  eqgen  13815  conjsubgen  13866  hmeoopn  15037  hmeocld  15038  hmeontr  15039  hmeoimaf1o  15040  usgrf1  16028  uspgr2wlkeq  16218  trlres  16243  iswomninnlem  16656