Theorem f1f1orn 5535

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 5485	. 2 |- ( F : A -1-1-> B -> F Fn A )
2		df-f1 5277	. . 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 5534	. 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 4675   ran crn 4677   Fun wfun 5266   Fn wfn 5267   -->wf 5268   -1-1->wf1 5269   -1-1-onto->wf1o 5271

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-in 3172  df-ss 3179  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279

This theorem is referenced by:  f1ores  5539  f1cnv  5548  f1cocnv1  5554  f1ocnvfvrneq  5853  ssenen  6950  f1dmvrnfibi  7048  cc2lem  7380  4sqlem11  12757  xpsff1o2  13216  imasmndf1  13319  imasgrpf1  13481  conjsubgen  13647  imasrngf1  13752  imasringf1  13860