Theorem fof 5106

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 2997	. . 3 |- ( ran F = B -> ran F C_ B )
2	1	anim2i 324	. 2 |- ( ( F Fn A /\ ran F = B ) -> ( F Fn A /\ ran F C_ B ) )
3		df-fo 4908	. 2 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
4		df-f 4906	. 2 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
5	2, 3, 4	3imtr4i 190	1 |- ( F : A -onto-> B -> F : A --> B )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   C_ wss 2917   ran crn 4346   Fn wfn 4897   -->wf 4898   -onto->wfo 4900

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931  df-f 4906  df-fo 4908

This theorem is referenced by:  fofun  5107  fofn  5108  dffo2  5110  foima  5111  resdif  5148  ffoss  5158  fconstfvm  5379  cocan2  5428  foeqcnvco  5430  fornex  5742  algrflem  5850  algrflemg  5851  tposf2  5883  ssdomg  6258  fopwdom  6310