Theorem f1f1orn 5453

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 5405	. 2 |- ( F : A -1-1-> B -> F Fn A )
2		df-f1 5203	. . 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 5452	. 2 |- ( F : A -1-1-onto-> ran F <-> ( F Fn A /\ Fun `' F ) )
5	1, 3, 4	sylanbrc 415	1 |- ( F : A -1-1-> B -> F : A -1-1-onto-> ran F )

Syntax hints:   -> wi 4   `'ccnv 4610   ran crn 4612   Fun wfun 5192   Fn wfn 5193   -->wf 5194   -1-1->wf1 5195   -1-1-onto->wf1o 5197

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-in 3127  df-ss 3134  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205

This theorem is referenced by:  f1ores  5457  f1cnv  5466  f1cocnv1  5472  f1ocnvfvrneq  5761  ssenen  6829  f1dmvrnfibi  6921  cc2lem  7228