Theorem dffo2 5357

Description: Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.)

Assertion

Ref	Expression
dffo2	|- ( F : A -onto-> B <-> ( F : A --> B /\ ran F = B ) )

Proof of Theorem dffo2

Step	Hyp	Ref	Expression
1		fof 5353	. . 3 |- ( F : A -onto-> B -> F : A --> B )
2		forn 5356	. . 3 |- ( F : A -onto-> B -> ran F = B )
3	1, 2	jca 304	. 2 |- ( F : A -onto-> B -> ( F : A --> B /\ ran F = B ) )
4		ffn 5280	. . 3 |- ( F : A --> B -> F Fn A )
5		df-fo 5137	. . . 4 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
6	5	biimpri 132	. . 3 |- ( ( F Fn A /\ ran F = B ) -> F : A -onto-> B )
7	4, 6	sylan 281	. 2 |- ( ( F : A --> B /\ ran F = B ) -> F : A -onto-> B )
8	3, 7	impbii 125	1 |- ( F : A -onto-> B <-> ( F : A --> B /\ ran F = B ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1332   ran crn 4548   Fn wfn 5126   -->wf 5127   -onto->wfo 5129

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-fo 5137

This theorem is referenced by:  foco  5363  dff1o5  5384  dffo3  5575  dffo4  5576  fo1stresm  6067  fo2ndresm  6068  fo2ndf  6132  1fv  9947