Theorem dff1o2 5372

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 5130	. 2 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
2		df-f1 5128	. . . 4 |- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
3		df-fo 5129	. . . 4 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
4	2, 3	anbi12i 455	. . 3 |- ( ( F : A -1-1-> B /\ F : A -onto-> B ) <-> ( ( F : A --> B /\ Fun `' F ) /\ ( F Fn A /\ ran F = B ) ) )
5		anass 398	. . . 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 974	. . . . . 6 |- ( ( F Fn A /\ Fun `' F /\ ran F = B ) <-> ( Fun `' F /\ ( F Fn A /\ ran F = B ) ) )
7	6	anbi1i 453	. . . . 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 3151	. . . . . . . 8 |- ( ran F = B -> ran F C_ B )
9		df-f 5127	. . . . . . . . 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 1000	. . . . . 6 |- ( ( F Fn A /\ Fun `' F /\ ran F = B ) -> F : A --> B )
13	12	pm4.71i 388	. . . . 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 962   = wceq 1331   C_ wss 3071   `'ccnv 4538   ran crn 4540   Fun wfun 5117   Fn wfn 5118   -->wf 5119   -1-1->wf1 5120   -onto->wfo 5121   -1-1-onto->wf1o 5122

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130

This theorem is referenced by:  dff1o3  5373  dff1o4  5375  f1orn  5377  dif1en  6773  fiintim  6817