Theorem f1f1orn 4649

Description: A one-to-one function maps one-to-one onto its range.

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		f1orn 4648	. 2 |- (F:A-1-1-onto->ran F <-> (F Fn A /\ Fun `'F))
2		f1f 4610	. . 3 |- (F:A-1-1->B -> F:A-->B)
3		ffn 4562	. . 3 |- (F:A-->B -> F Fn A)
4	2, 3	syl 12	. 2 |- (F:A-1-1->B -> F Fn A)
5		df-f1 4011	. . 3 |- (F:A-1-1->B <-> (F:A-->B /\ Fun `'F))
6	5	simprbi 353	. 2 |- (F:A-1-1->B -> Fun `'F)
7	1, 4, 6	sylanbrc 527	1 |- (F:A-1-1->B -> F:A-1-1-onto->ran F)

Syntax hints:   -> wi 3  `'ccnv 3985  ran crn 3987  Fun wfun 3992   Fn wfn 3993  -->wf 3994  -1-1->wf1 3995  -1-1-onto->wf1o 3997

This theorem is referenced by:  f1dmex 4664  cnvinj 14463  cmpinj 14464  cmpinj2 14465  enf1f1o 15720

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-in 2603  df-ss 2605  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013

Copyright terms: Public domain