Theorem f1f1orn 5515

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 5465	. 2 |- ( F : A -1-1-> B -> F Fn A )
2		df-f1 5263	. . 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 5514	. 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 4662   ran crn 4664   Fun wfun 5252   Fn wfn 5253   -->wf 5254   -1-1->wf1 5255   -1-1-onto->wf1o 5257

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-in 3163  df-ss 3170  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265

This theorem is referenced by:  f1ores  5519  f1cnv  5528  f1cocnv1  5534  f1ocnvfvrneq  5829  ssenen  6912  f1dmvrnfibi  7010  cc2lem  7333  4sqlem11  12570  xpsff1o2  12994  imasgrpf1  13242  conjsubgen  13408  imasrngf1  13513  imasringf1  13621