Theorem dff1o5 5553

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 5297	. 2 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
2		dffo2 5524	. . . 4 |- ( F : A -onto-> B <-> ( F : A --> B /\ ran F = B ) )
3		f1f 5503	. . . . 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 1373   ran crn 4694   -->wf 5286   -1-1->wf1 5287   -onto->wfo 5288   -1-1-onto->wf1o 5289

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-in 3180  df-ss 3187  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297

This theorem is referenced by:  f1orescnv  5560  f1finf1o  7075  djuinr  7191  eninl  7225  eninr  7226  frec2uzf1od  10588  ennnfonelemex  12900  ennnfonelemen  12907  ssnnctlemct  12932  2lgslem1b  15681  pwf1oexmid  16138