Theorem dff1o2 5151

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 4929	. 2 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
2		df-f1 4927	. . . 4 |- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
3		df-fo 4928	. . . 4 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
4	2, 3	anbi12i 447	. . 3 |- ( ( F : A -1-1-> B /\ F : A -onto-> B ) <-> ( ( F : A --> B /\ Fun `' F ) /\ ( F Fn A /\ ran F = B ) ) )
5		anass 393	. . . 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 931	. . . . . 6 |- ( ( F Fn A /\ Fun `' F /\ ran F = B ) <-> ( Fun `' F /\ ( F Fn A /\ ran F = B ) ) )
7	6	anbi1i 445	. . . . 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 3051	. . . . . . . 8 |- ( ran F = B -> ran F C_ B )
9		df-f 4926	. . . . . . . . 9 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
10	9	biimpri 131	. . . . . . . 8 |- ( ( F Fn A /\ ran F C_ B ) -> F : A --> B )
11	8, 10	sylan2 280	. . . . . . 7 |- ( ( F Fn A /\ ran F = B ) -> F : A --> B )
12	11	3adant2 957	. . . . . 6 |- ( ( F Fn A /\ Fun `' F /\ ran F = B ) -> F : A --> B )
13	12	pm4.71i 383	. . . . 5 |- ( ( F Fn A /\ Fun `' F /\ ran F = B ) <-> ( ( F Fn A /\ Fun `' F /\ ran F = B ) /\ F : A --> B ) )
14		ancom 262	. . . . 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 211	. . . 4 |- ( ( F : A --> B /\ ( Fun `' F /\ ( F Fn A /\ ran F = B ) ) ) <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )
16	5, 15	bitri 182	. . 3 |- ( ( ( F : A --> B /\ Fun `' F ) /\ ( F Fn A /\ ran F = B ) ) <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )
17	4, 16	bitri 182	. 2 |- ( ( F : A -1-1-> B /\ F : A -onto-> B ) <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )
18	1, 17	bitri 182	1 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )

Syntax hints:   /\ wa 102   <-> wb 103   /\ w3a 919   = wceq 1284   C_ wss 2973   `'ccnv 4362   ran crn 4364   Fun wfun 4916   Fn wfn 4917   -->wf 4918   -1-1->wf1 4919   -onto->wfo 4920   -1-1-onto->wf1o 4921

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-in 2979  df-ss 2986  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929

This theorem is referenced by:  dff1o3  5152  dff1o4  5154  f1orn  5156  dif1en  6364