Theorem dffo2 5110

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 5106	. . 3 |- ( F : A -onto-> B -> F : A --> B )
2		forn 5109	. . 3 |- ( F : A -onto-> B -> ran F = B )
3	1, 2	jca 290	. 2 |- ( F : A -onto-> B -> ( F : A --> B /\ ran F = B ) )
4		ffn 5046	. . 3 |- ( F : A --> B -> F Fn A )
5		df-fo 4908	. . . 4 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
6	5	biimpri 124	. . 3 |- ( ( F Fn A /\ ran F = B ) -> F : A -onto-> B )
7	4, 6	sylan 267	. 2 |- ( ( F : A --> B /\ ran F = B ) -> F : A -onto-> B )
8	3, 7	impbii 117	1 |- ( F : A -onto-> B <-> ( F : A --> B /\ ran F = B ) )

Syntax hints:   /\ wa 97   <-> wb 98   = wceq 1243   ran crn 4346   Fn wfn 4897   -->wf 4898   -onto->wfo 4900

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931  df-f 4906  df-fo 4908

This theorem is referenced by:  foco  5116  dff1o5  5135  dffo3  5314  dffo4  5315  fo1stresm  5788  fo2ndresm  5789  fo2ndf  5848  1fv  8996