Theorem dffo2 5619

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 5615	. . 3 |- ( F : A -onto-> B -> F : A --> B )
2		forn 5618	. . 3 |- ( F : A -onto-> B -> ran F = B )
3	1, 2	jca 306	. 2 |- ( F : A -onto-> B -> ( F : A --> B /\ ran F = B ) )
4		ffn 5533	. . 3 |- ( F : A --> B -> F Fn A )
5		df-fo 5383	. . . 4 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
6	5	biimpri 133	. . 3 |- ( ( F Fn A /\ ran F = B ) -> F : A -onto-> B )
7	4, 6	sylan 283	. 2 |- ( ( F : A --> B /\ ran F = B ) -> F : A -onto-> B )
8	3, 7	impbii 126	1 |- ( F : A -onto-> B <-> ( F : A --> B /\ ran F = B ) )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   = wceq 1402   ran crn 4775   Fn wfn 5372   -->wf 5373   -onto->wfo 5375

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233  df-f 5381  df-fo 5383

This theorem is used by:  foco  5626  dff1o5  5648  dffo3  5855  dffo4  5856  fo1stresm  6395  fo2ndresm  6396  fo2ndf  6463  1fv  10546