Theorem f1f1orn 5474

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 5425	. 2 |- ( F : A -1-1-> B -> F Fn A )
2		df-f1 5223	. . 3 |- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
3	2	simprbi 275	. 2 |- ( F : A -1-1-> B -> Fun `' F )
4		f1orn 5473	. 2 |- ( F : A -1-1-onto-> ran F <-> ( F Fn A /\ Fun `' F ) )
5	1, 3, 4	sylanbrc 417	1 |- ( F : A -1-1-> B -> F : A -1-1-onto-> ran F )

Syntax hints:   -> wi 4   `'ccnv 4627   ran crn 4629   Fun wfun 5212   Fn wfn 5213   -->wf 5214   -1-1->wf1 5215   -1-1-onto->wf1o 5217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3137  df-ss 3144  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225

This theorem is referenced by:  f1ores  5478  f1cnv  5487  f1cocnv1  5493  f1ocnvfvrneq  5785  ssenen  6853  f1dmvrnfibi  6945  cc2lem  7267  xpsff1o2  12775