Theorem dff1o2 5485

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 5242	. 2 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
2		df-f1 5240	. . . 4 |- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
3		df-fo 5241	. . . 4 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
4	2, 3	anbi12i 460	. . 3 |- ( ( F : A -1-1-> B /\ F : A -onto-> B ) <-> ( ( F : A --> B /\ Fun `' F ) /\ ( F Fn A /\ ran F = B ) ) )
5		anass 401	. . . 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 992	. . . . . 6 |- ( ( F Fn A /\ Fun `' F /\ ran F = B ) <-> ( Fun `' F /\ ( F Fn A /\ ran F = B ) ) )
7	6	anbi1i 458	. . . . 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 3224	. . . . . . . 8 |- ( ran F = B -> ran F C_ B )
9		df-f 5239	. . . . . . . . 9 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
10	9	biimpri 133	. . . . . . . 8 |- ( ( F Fn A /\ ran F C_ B ) -> F : A --> B )
11	8, 10	sylan2 286	. . . . . . 7 |- ( ( F Fn A /\ ran F = B ) -> F : A --> B )
12	11	3adant2 1018	. . . . . 6 |- ( ( F Fn A /\ Fun `' F /\ ran F = B ) -> F : A --> B )
13	12	pm4.71i 391	. . . . 5 |- ( ( F Fn A /\ Fun `' F /\ ran F = B ) <-> ( ( F Fn A /\ Fun `' F /\ ran F = B ) /\ F : A --> B ) )
14		ancom 266	. . . . 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 213	. . . 4 |- ( ( F : A --> B /\ ( Fun `' F /\ ( F Fn A /\ ran F = B ) ) ) <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )
16	5, 15	bitri 184	. . 3 |- ( ( ( F : A --> B /\ Fun `' F ) /\ ( F Fn A /\ ran F = B ) ) <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )
17	4, 16	bitri 184	. 2 |- ( ( F : A -1-1-> B /\ F : A -onto-> B ) <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )
18	1, 17	bitri 184	1 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   C_ wss 3144   `'ccnv 4643   ran crn 4645   Fun wfun 5229   Fn wfn 5230   -->wf 5231   -1-1->wf1 5232   -onto->wfo 5233   -1-1-onto->wf1o 5234

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-in 3150  df-ss 3157  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242

This theorem is referenced by:  dff1o3  5486  dff1o4  5488  f1orn  5490  dif1en  6908  fiintim  6958