Theorem dffo2 5414

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 5410	. . 3 |- ( F : A -onto-> B -> F : A --> B )
2		forn 5413	. . 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 5337	. . 3 |- ( F : A --> B -> F Fn A )
5		df-fo 5194	. . . 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 1343   ran crn 4605   Fn wfn 5183   -->wf 5184   -onto->wfo 5186

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3122  df-ss 3129  df-f 5192  df-fo 5194

This theorem is referenced by:  foco  5420  dff1o5  5441  dffo3  5632  dffo4  5633  fo1stresm  6129  fo2ndresm  6130  fo2ndf  6195  1fv  10074