Theorem dff1o5 5510

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 5262	. 2 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
2		dffo2 5481	. . . 4 |- ( F : A -onto-> B <-> ( F : A --> B /\ ran F = B ) )
3		f1f 5460	. . . . 5 |- ( F : A -1-1-> B -> F : A --> B )
4	3	biantrurd 305	. . . 4 |- ( F : A -1-1-> B -> ( ran F = B <-> ( F : A --> B /\ ran F = B ) ) )
5	2, 4	bitr4id 199	. . 3 |- ( F : A -1-1-> B -> ( F : A -onto-> B <-> ran F = B ) )
6	5	pm5.32i 454	. 2 |- ( ( F : A -1-1-> B /\ F : A -onto-> B ) <-> ( F : A -1-1-> B /\ ran F = B ) )
7	1, 6	bitri 184	1 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ ran F = B ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1364   ran crn 4661   -->wf 5251   -1-1->wf1 5252   -onto->wfo 5253   -1-1-onto->wf1o 5254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-in 3160  df-ss 3167  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262

This theorem is referenced by:  f1orescnv  5517  f1finf1o  7008  djuinr  7124  eninl  7158  eninr  7159  frec2uzf1od  10480  ennnfonelemex  12574  ennnfonelemen  12581  ssnnctlemct  12606  2lgslem1b  15246  pwf1oexmid  15560