Theorem dffo2 5130

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 5126	. . 3 |- ( F : A -onto-> B -> F : A --> B )
2		forn 5129	. . 3 |- ( F : A -onto-> B -> ran F = B )
3	1, 2	jca 300	. 2 |- ( F : A -onto-> B -> ( F : A --> B /\ ran F = B ) )
4		ffn 5066	. . 3 |- ( F : A --> B -> F Fn A )
5		df-fo 4928	. . . 4 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
6	5	biimpri 131	. . 3 |- ( ( F Fn A /\ ran F = B ) -> F : A -onto-> B )
7	4, 6	sylan 277	. 2 |- ( ( F : A --> B /\ ran F = B ) -> F : A -onto-> B )
8	3, 7	impbii 124	1 |- ( F : A -onto-> B <-> ( F : A --> B /\ ran F = B ) )

Syntax hints:   /\ wa 102   <-> wb 103   = wceq 1284   ran crn 4364   Fn wfn 4917   -->wf 4918   -onto->wfo 4920

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-in 2979  df-ss 2986  df-f 4926  df-fo 4928

This theorem is referenced by:  foco  5136  dff1o5  5155  dffo3  5335  dffo4  5336  fo1stresm  5808  fo2ndresm  5809  fo2ndf  5868  1fv  9149