Theorem f1f 5551

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 5338	. 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 4730   Fun wfun 5327   -->wf 5329   -1-1->wf1 5330

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 5338

This theorem is referenced by:  f1rn  5552  f1fn  5553  f1ss  5557  f1ssres  5560  f1of  5592  dff1o5  5601  fsnd  5637  cocan1  5938  f1o2ndf1  6402  brdomg  6962  f1dom2g  6972  f1domg  6974  dom3d  6990  f1imaen2g  7010  2dom  7023  1dom1el  7036  dom1o  7045  xpdom2  7058  dom0  7067  phplem4dom  7091  isinfinf  7129  infm  7139  updjudhcoinlf  7339  updjudhcoinrg  7340  casef1  7349  djudom  7352  difinfsnlem  7358  difinfsn  7359  seqf1oglem1  10844  fihashf1rn  11113  ennnfonelemrn  13120  reeff1o  15584  ushgruhgr  16021  umgr0e  16059  usgredgssen  16103  ausgrusgrben  16109  usgrss  16118  uspgrupgr  16122  usgrumgr  16125  usgrislfuspgrdom  16131  ushgredgedg  16167  ushgredgedgloop  16169  trlsegvdeglem6  16406  trlsegvdeglem7  16407  trlsegvdegfi  16408  eupth2lem3lem2fi  16410  eupth2lem3lem3fi  16411  eupth2lem3lem6fi  16412  eupth2lem3lem4fi  16414  3dom  16708  pwle2  16720