Theorem fof 5126

Description: An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)

Assertion

Ref	Expression
fof	|- ( F : A -onto-> B -> F : A --> B )

Proof of Theorem fof

Step	Hyp	Ref	Expression
1		eqimss 3051	. . 3 |- ( ran F = B -> ran F C_ B )
2	1	anim2i 334	. 2 |- ( ( F Fn A /\ ran F = B ) -> ( F Fn A /\ ran F C_ B ) )
3		df-fo 4928	. 2 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
4		df-f 4926	. 2 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
5	2, 3, 4	3imtr4i 199	1 |- ( F : A -onto-> B -> F : A --> B )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   C_ wss 2973   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:  fofun  5127  fofn  5128  dffo2  5130  foima  5131  resdif  5168  ffoss  5178  fconstfvm  5400  cocan2  5448  foeqcnvco  5450  fornex  5762  algrflem  5870  algrflemg  5871  tposf2  5906  ssdomg  6281  fopwdom  6333