Theorem f1f 5578

Description: A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)

Assertion

Ref	Expression
f1f	|- ( F : A -1-1-> B -> F : A --> B )

Proof of Theorem f1f

Step	Hyp	Ref	Expression
1		df-f1 5362	. 2 |- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
2	1	simplbi 274	1 |- ( F : A -1-1-> B -> F : A --> B )

Syntax hints:   -> wi 4   `'ccnv 4753   Fun wfun 5351   -->wf 5353   -1-1->wf1 5354

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-f1 5362

This theorem is referenced by:  f1rn  5579  f1fn  5580  f1ss  5584  f1ssres  5587  f1of  5619  dff1o5  5628  fsnd  5664  cocan1  5966  f1o2ndf1  6437  brdomg  6998  f1dom2g  7008  f1domg  7010  dom3d  7026  f1imaen2g  7046  2dom  7059  1dom1el  7073  dom1o  7082  xpdom2  7095  dom0  7104  phplem4dom  7129  isinfinf  7167  infm  7177  updjudhcoinlf  7384  updjudhcoinrg  7385  casef1  7394  djudom  7397  difinfsnlem  7403  difinfsn  7404  seqf1oglem1  10905  fihashf1rn  11176  ennnfonelemrn  13254  reeff1o  15764  ushgruhgr  16201  umgr0e  16239  usgredgssen  16283  ausgrusgrben  16289  usgrss  16298  uspgrupgr  16302  usgrumgr  16305  usgrislfuspgrdom  16311  ushgredgedg  16347  ushgredgedgloop  16349  trlsegvdeglem6  16586  trlsegvdeglem7  16587  trlsegvdegfi  16588  eupth2lem3lem2fi  16590  eupth2lem3lem3fi  16591  eupth2lem3lem6fi  16592  eupth2lem3lem4fi  16594  3dom  16888  pwle2  16898