Theorem dff1o2 6142

Description: Alternate definition of one-to-one onto function. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Assertion

Ref	Expression
dff1o2	|- ( F : A -1-1-onto-> B <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )

Proof of Theorem dff1o2

Step	Hyp	Ref	Expression
1		df-f1o 5895	. 2 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
2		df-f1 5893	. . . 4 |- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
3		df-fo 5894	. . . 4 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
4	2, 3	anbi12i 733	. . 3 |- ( ( F : A -1-1-> B /\ F : A -onto-> B ) <-> ( ( F : A --> B /\ Fun `' F ) /\ ( F Fn A /\ ran F = B ) ) )
5		anass 681	. . . 4 |- ( ( ( F : A --> B /\ Fun `' F ) /\ ( F Fn A /\ ran F = B ) ) <-> ( F : A --> B /\ ( Fun `' F /\ ( F Fn A /\ ran F = B ) ) ) )
6		3anan12 1051	. . . . . 6 |- ( ( F Fn A /\ Fun `' F /\ ran F = B ) <-> ( Fun `' F /\ ( F Fn A /\ ran F = B ) ) )
7	6	anbi1i 731	. . . . 5 |- ( ( ( F Fn A /\ Fun `' F /\ ran F = B ) /\ F : A --> B ) <-> ( ( Fun `' F /\ ( F Fn A /\ ran F = B ) ) /\ F : A --> B ) )
8		eqimss 3657	. . . . . . . 8 |- ( ran F = B -> ran F C_ B )
9		df-f 5892	. . . . . . . . 9 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
10	9	biimpri 218	. . . . . . . 8 |- ( ( F Fn A /\ ran F C_ B ) -> F : A --> B )
11	8, 10	sylan2 491	. . . . . . 7 |- ( ( F Fn A /\ ran F = B ) -> F : A --> B )
12	11	3adant2 1080	. . . . . 6 |- ( ( F Fn A /\ Fun `' F /\ ran F = B ) -> F : A --> B )
13	12	pm4.71i 664	. . . . 5 |- ( ( F Fn A /\ Fun `' F /\ ran F = B ) <-> ( ( F Fn A /\ Fun `' F /\ ran F = B ) /\ F : A --> B ) )
14		ancom 466	. . . . 5 |- ( ( F : A --> B /\ ( Fun `' F /\ ( F Fn A /\ ran F = B ) ) ) <-> ( ( Fun `' F /\ ( F Fn A /\ ran F = B ) ) /\ F : A --> B ) )
15	7, 13, 14	3bitr4ri 293	. . . 4 |- ( ( F : A --> B /\ ( Fun `' F /\ ( F Fn A /\ ran F = B ) ) ) <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )
16	5, 15	bitri 264	. . 3 |- ( ( ( F : A --> B /\ Fun `' F ) /\ ( F Fn A /\ ran F = B ) ) <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )
17	4, 16	bitri 264	. 2 |- ( ( F : A -1-1-> B /\ F : A -onto-> B ) <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )
18	1, 17	bitri 264	1 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   C_ wss 3574   `'ccnv 5113   ran crn 5115   Fun wfun 5882   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-in 3581  df-ss 3588  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895

This theorem is referenced by:  dff1o3  6143  dff1o4  6145  f1orn  6147  tz7.49c  7541  fiint  8237  symgfixelsi  17855  dfrelog  24312  adj1o  28753  fresf1o  29433  f1mptrn  29435  esumc  30113  ntrneinex  38375  stoweidlem39  40256