Theorem dff1o2 5437

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 5195	. 2 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
2		df-f1 5193	. . . 4 |- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
3		df-fo 5194	. . . 4 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
4	2, 3	anbi12i 456	. . 3 |- ( ( F : A -1-1-> B /\ F : A -onto-> B ) <-> ( ( F : A --> B /\ Fun `' F ) /\ ( F Fn A /\ ran F = B ) ) )
5		anass 399	. . . 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 980	. . . . . 6 |- ( ( F Fn A /\ Fun `' F /\ ran F = B ) <-> ( Fun `' F /\ ( F Fn A /\ ran F = B ) ) )
7	6	anbi1i 454	. . . . 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 3196	. . . . . . . 8 |- ( ran F = B -> ran F C_ B )
9		df-f 5192	. . . . . . . . 9 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
10	9	biimpri 132	. . . . . . . 8 |- ( ( F Fn A /\ ran F C_ B ) -> F : A --> B )
11	8, 10	sylan2 284	. . . . . . 7 |- ( ( F Fn A /\ ran F = B ) -> F : A --> B )
12	11	3adant2 1006	. . . . . 6 |- ( ( F Fn A /\ Fun `' F /\ ran F = B ) -> F : A --> B )
13	12	pm4.71i 389	. . . . 5 |- ( ( F Fn A /\ Fun `' F /\ ran F = B ) <-> ( ( F Fn A /\ Fun `' F /\ ran F = B ) /\ F : A --> B ) )
14		ancom 264	. . . . 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 212	. . . 4 |- ( ( F : A --> B /\ ( Fun `' F /\ ( F Fn A /\ ran F = B ) ) ) <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )
16	5, 15	bitri 183	. . 3 |- ( ( ( F : A --> B /\ Fun `' F ) /\ ( F Fn A /\ ran F = B ) ) <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )
17	4, 16	bitri 183	. 2 |- ( ( F : A -1-1-> B /\ F : A -onto-> B ) <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )
18	1, 17	bitri 183	1 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )

Syntax hints:   /\ wa 103   <-> wb 104   /\ w3a 968   = wceq 1343   C_ wss 3116   `'ccnv 4603   ran crn 4605   Fun wfun 5182   Fn wfn 5183   -->wf 5184   -1-1->wf1 5185   -onto->wfo 5186   -1-1-onto->wf1o 5187

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3122  df-ss 3129  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195

This theorem is referenced by:  dff1o3  5438  dff1o4  5440  f1orn  5442  dif1en  6845  fiintim  6894