Theorem f1f1orn 5371

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 5325	. 2 |- ( F : A -1-1-> B -> F Fn A )
2		df-f1 5123	. . 3 |- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
3	2	simprbi 273	. 2 |- ( F : A -1-1-> B -> Fun `' F )
4		f1orn 5370	. 2 |- ( F : A -1-1-onto-> ran F <-> ( F Fn A /\ Fun `' F ) )
5	1, 3, 4	sylanbrc 413	1 |- ( F : A -1-1-> B -> F : A -1-1-onto-> ran F )

Syntax hints:   -> wi 4   `'ccnv 4533   ran crn 4535   Fun wfun 5112   Fn wfn 5113   -->wf 5114   -1-1->wf1 5115   -1-1-onto->wf1o 5117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-in 3072  df-ss 3079  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125

This theorem is referenced by:  f1ores  5375  f1cnv  5384  f1cocnv1  5390  f1ocnvfvrneq  5676  ssenen  6738  f1dmvrnfibi  6825