Theorem dffo2 5424

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 5420	. . 3 |- ( F : A -onto-> B -> F : A --> B )
2		forn 5423	. . 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 5347	. . 3 |- ( F : A --> B -> F Fn A )
5		df-fo 5204	. . . 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 1348   ran crn 4612   Fn wfn 5193   -->wf 5194   -onto->wfo 5196

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-in 3127  df-ss 3134  df-f 5202  df-fo 5204

This theorem is referenced by:  foco  5430  dff1o5  5451  dffo3  5643  dffo4  5644  fo1stresm  6140  fo2ndresm  6141  fo2ndf  6206  1fv  10095