Theorem f1f1orn 5555

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 5505	. 2 |- ( F : A -1-1-> B -> F Fn A )
2		df-f1 5295	. . 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 5554	. 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 4692   ran crn 4694   Fun wfun 5284   Fn wfn 5285   -->wf 5286   -1-1->wf1 5287   -1-1-onto->wf1o 5289

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-in 3180  df-ss 3187  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297

This theorem is referenced by:  f1ores  5559  f1cnv  5568  f1cocnv1  5574  f1ocnvfvrneq  5874  ssenen  6973  f1dmvrnfibi  7072  cc2lem  7413  4sqlem11  12839  xpsff1o2  13298  imasmndf1  13401  imasgrpf1  13563  conjsubgen  13729  imasrngf1  13834  imasringf1  13942