Theorem dff1o5 5440

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 5194	. 2 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
2		dffo2 5413	. . . 4 |- ( F : A -onto-> B <-> ( F : A --> B /\ ran F = B ) )
3		f1f 5392	. . . . 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 1343   ran crn 4604   -->wf 5183   -1-1->wf1 5184   -onto->wfo 5185   -1-1-onto->wf1o 5186

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-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3121  df-ss 3128  df-f 5191  df-f1 5192  df-fo 5193  df-f1o 5194

This theorem is referenced by:  f1orescnv  5447  f1finf1o  6908  djuinr  7024  eninl  7058  eninr  7059  frec2uzf1od  10337  ennnfonelemex  12343  ennnfonelemen  12350  ssnnctlemct  12375  pwf1oexmid  13839