Theorem fof 5233

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 3078	. . 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 5021	. 2 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
4		df-f 5019	. 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 1289   C_ wss 2999   ran crn 4439   Fn wfn 5010   -->wf 5011   -onto->wfo 5013

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 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3005  df-ss 3012  df-f 5019  df-fo 5021

This theorem is referenced by:  fofun  5234  fofn  5235  dffo2  5237  foima  5238  resdif  5275  ffoss  5285  fconstfvm  5515  cocan2  5567  foeqcnvco  5569  fornex  5886  algrflem  5994  algrflemg  5995  tposf2  6033  mapsn  6445  ssdomg  6493  fopwdom  6550  fidcenumlemrks  6660  fidcenumlemr  6662  fodjuomnilemdc  6797  exmidfodomrlemr  6826  exmidfodomrlemrALT  6827  focdmex  10191