Theorem dff1o5 5369

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 5125	. 2 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
2		f1f 5323	. . . . 5 |- ( F : A -1-1-> B -> F : A --> B )
3	2	biantrurd 303	. . . 4 |- ( F : A -1-1-> B -> ( ran F = B <-> ( F : A --> B /\ ran F = B ) ) )
4		dffo2 5344	. . . 4 |- ( F : A -onto-> B <-> ( F : A --> B /\ ran F = B ) )
5	3, 4	syl6rbbr 198	. . 3 |- ( F : A -1-1-> B -> ( F : A -onto-> B <-> ran F = B ) )
6	5	pm5.32i 449	. 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 1331   ran crn 4535   -->wf 5114   -1-1->wf1 5115   -onto->wfo 5116   -1-1-onto->wf1o 5117

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 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-in 3072  df-ss 3079  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125

This theorem is referenced by:  f1orescnv  5376  f1finf1o  6828  djuinr  6941  eninl  6975  eninr  6976  frec2uzf1od  10172  ennnfonelemex  11916  ennnfonelemen  11923  pwf1oexmid  13183