Theorem dff1o5 5384

Description: Alternate definition of one-to-one onto function. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Assertion

Ref	Expression
dff1o5	|- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ ran F = B ) )

Proof of Theorem dff1o5

Step	Hyp	Ref	Expression
1		df-f1o 5138	. 2 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
2		dffo2 5357	. . . 4 |- ( F : A -onto-> B <-> ( F : A --> B /\ ran F = B ) )
3		f1f 5336	. . . . 5 |- ( F : A -1-1-> B -> F : A --> B )
4	3	biantrurd 303	. . . 4 |- ( F : A -1-1-> B -> ( ran F = B <-> ( F : A --> B /\ ran F = B ) ) )
5	2, 4	bitr4id 198	. . 3 |- ( F : A -1-1-> B -> ( F : A -onto-> B <-> ran F = B ) )
6	5	pm5.32i 450	. 2 |- ( ( F : A -1-1-> B /\ F : A -onto-> B ) <-> ( F : A -1-1-> B /\ ran F = B ) )
7	1, 6	bitri 183	1 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ ran F = B ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1332   ran crn 4548   -->wf 5127   -1-1->wf1 5128   -onto->wfo 5129   -1-1-onto->wf1o 5130

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-in 3082  df-ss 3089  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138

This theorem is referenced by:  f1orescnv  5391  f1finf1o  6843  djuinr  6956  eninl  6990  eninr  6991  frec2uzf1od  10210  ennnfonelemex  11963  ennnfonelemen  11970  pwf1oexmid  13367