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Theorem foconst 5656
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 5586 . . . . 5  |-  ( F : A --> { B }  ->  Rel  F )
2 relrn0 5120 . . . . . 6  |-  ( Rel 
F  ->  ( F  =  (/)  <->  ran  F  =  (/) ) )
32necon3abid 2631 . . . . 5  |-  ( Rel 
F  ->  ( F  =/=  (/)  <->  -.  ran  F  =  (/) ) )
41, 3syl 16 . . . 4  |-  ( F : A --> { B }  ->  ( F  =/=  (/) 
<->  -.  ran  F  =  (/) ) )
5 frn 5589 . . . . . 6  |-  ( F : A --> { B }  ->  ran  F  C_  { B } )
6 sssn 3949 . . . . . 6  |-  ( ran 
F  C_  { B } 
<->  ( ran  F  =  (/)  \/  ran  F  =  { B } ) )
75, 6sylib 189 . . . . 5  |-  ( F : A --> { B }  ->  ( ran  F  =  (/)  \/  ran  F  =  { B } ) )
87ord 367 . . . 4  |-  ( F : A --> { B }  ->  ( -.  ran  F  =  (/)  ->  ran  F  =  { B } ) )
94, 8sylbid 207 . . 3  |-  ( F : A --> { B }  ->  ( F  =/=  (/)  ->  ran  F  =  { B } ) )
109imdistani 672 . 2  |-  ( ( F : A --> { B }  /\  F  =/=  (/) )  -> 
( F : A --> { B }  /\  ran  F  =  { B }
) )
11 dffo2 5649 . 2  |-  ( F : A -onto-> { B } 
<->  ( F : A --> { B }  /\  ran  F  =  { B }
) )
1210, 11sylibr 204 1  |-  ( ( F : A --> { B }  /\  F  =/=  (/) )  ->  F : A -onto-> { B } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    =/= wne 2598    C_ wss 3312   (/)c0 3620   {csn 3806   ran crn 4871   Rel wrel 4875   -->wf 5442   -onto->wfo 5444
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-dm 4880  df-rn 4881  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452
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