Theorem f1f1orn 5157

Description: A one-to-one function maps one-to-one onto its range. (Contributed by NM, 4-Sep-2004.)

Assertion

Ref	Expression
f1f1orn	|- ( F : A -1-1-> B -> F : A -1-1-onto-> ran F )

Proof of Theorem f1f1orn

Step	Hyp	Ref	Expression
1		f1fn 5113	. 2 |- ( F : A -1-1-> B -> F Fn A )
2		df-f1 4927	. . 3 |- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
3	2	simprbi 269	. 2 |- ( F : A -1-1-> B -> Fun `' F )
4		f1orn 5156	. 2 |- ( F : A -1-1-onto-> ran F <-> ( F Fn A /\ Fun `' F ) )
5	1, 3, 4	sylanbrc 408	1 |- ( F : A -1-1-> B -> F : A -1-1-onto-> ran F )

Syntax hints:   -> wi 4   `'ccnv 4362   ran crn 4364   Fun wfun 4916   Fn wfn 4917   -->wf 4918   -1-1->wf1 4919   -1-1-onto->wf1o 4921

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-in 2979  df-ss 2986  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929

This theorem is referenced by:  f1ores  5161  f1cnv  5170  f1cocnv1  5176  f1ocnvfvrneq  5442  f1dmvrnfibi  6393