Theorem f1f1orn 5080

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 5036	. 2 |- ( F :A -1-1->B -> F Fn A)
2		df-f1 4850	. . 3 |- ( F :A -1-1->B <-> ( F :A -->B /\ Fun `' F))
3	2	simprbi 260	. 2 |- ( F :A -1-1->B -> Fun `' F)
4		f1orn 5079	. 2 |- ( F :A -1-1-onto-> ran F <-> ( F Fn A /\ Fun `' F))
5	1, 3, 4	sylanbrc 394	1 |- ( F :A -1-1->B -> F :A -1-1-onto-> ran F)

Syntax hints:   -> wi 4   `'ccnv 4287   ran crn 4289   Fun wfun 4839   Fn wfn 4840   -->wf 4841   -1-1->wf1 4842   -1-1-onto->wf1o 4844

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852

This theorem is referenced by:  f1ores  5084  f1cnv  5093  f1cocnv1  5099  f1ocnvfvrneq  5365