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Theorem foconst 5462
Description: A nonzero constant function is onto. (Contributed by NM, 12-Jan-2007.)
Assertion
Ref Expression
foconst  |-  ( ( F : A --> { B }  /\  F  =/=  (/) )  ->  F : A -onto-> { B } )

Proof of Theorem foconst
StepHypRef Expression
1 frel 5392 . . . . 5  |-  ( F : A --> { B }  ->  Rel  F )
2 relrn0 4937 . . . . . 6  |-  ( Rel 
F  ->  ( F  =  (/)  <->  ran  F  =  (/) ) )
32necon3abid 2479 . . . . 5  |-  ( Rel 
F  ->  ( F  =/=  (/)  <->  -.  ran  F  =  (/) ) )
41, 3syl 15 . . . 4  |-  ( F : A --> { B }  ->  ( F  =/=  (/) 
<->  -.  ran  F  =  (/) ) )
5 frn 5395 . . . . . 6  |-  ( F : A --> { B }  ->  ran  F  C_  { B } )
6 sssn 3772 . . . . . 6  |-  ( ran 
F  C_  { B } 
<->  ( ran  F  =  (/)  \/  ran  F  =  { B } ) )
75, 6sylib 188 . . . . 5  |-  ( F : A --> { B }  ->  ( ran  F  =  (/)  \/  ran  F  =  { B } ) )
87ord 366 . . . 4  |-  ( F : A --> { B }  ->  ( -.  ran  F  =  (/)  ->  ran  F  =  { B } ) )
94, 8sylbid 206 . . 3  |-  ( F : A --> { B }  ->  ( F  =/=  (/)  ->  ran  F  =  { B } ) )
109imdistani 671 . 2  |-  ( ( F : A --> { B }  /\  F  =/=  (/) )  -> 
( F : A --> { B }  /\  ran  F  =  { B }
) )
11 dffo2 5455 . 2  |-  ( F : A -onto-> { B } 
<->  ( F : A --> { B }  /\  ran  F  =  { B }
) )
1210, 11sylibr 203 1  |-  ( ( F : A --> { B }  /\  F  =/=  (/) )  ->  F : A -onto-> { B } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    =/= wne 2446    C_ wss 3152   (/)c0 3455   {csn 3640   ran crn 4690   Rel wrel 4694   -->wf 5251   -onto->wfo 5253
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261
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