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Theorem foconst 5731
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 5662 . . . . 5  |-  ( F : A --> { B }  ->  Rel  F )
2 relrn0 5197 . . . . . 6  |-  ( Rel 
F  ->  ( F  =  (/)  <->  ran  F  =  (/) ) )
32necon3abid 2694 . . . . 5  |-  ( Rel 
F  ->  ( F  =/=  (/)  <->  -.  ran  F  =  (/) ) )
41, 3syl 16 . . . 4  |-  ( F : A --> { B }  ->  ( F  =/=  (/) 
<->  -.  ran  F  =  (/) ) )
5 frn 5665 . . . . . 6  |-  ( F : A --> { B }  ->  ran  F  C_  { B } )
6 sssn 4131 . . . . . 6  |-  ( ran 
F  C_  { B } 
<->  ( ran  F  =  (/)  \/  ran  F  =  { B } ) )
75, 6sylib 196 . . . . 5  |-  ( F : A --> { B }  ->  ( ran  F  =  (/)  \/  ran  F  =  { B } ) )
87ord 377 . . . 4  |-  ( F : A --> { B }  ->  ( -.  ran  F  =  (/)  ->  ran  F  =  { B } ) )
94, 8sylbid 215 . . 3  |-  ( F : A --> { B }  ->  ( F  =/=  (/)  ->  ran  F  =  { B } ) )
109imdistani 690 . 2  |-  ( ( F : A --> { B }  /\  F  =/=  (/) )  -> 
( F : A --> { B }  /\  ran  F  =  { B }
) )
11 dffo2 5724 . 2  |-  ( F : A -onto-> { B } 
<->  ( F : A --> { B }  /\  ran  F  =  { B }
) )
1210, 11sylibr 212 1  |-  ( ( F : A --> { B }  /\  F  =/=  (/) )  ->  F : A -onto-> { B } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370    =/= wne 2644    C_ wss 3428   (/)c0 3737   {csn 3977   ran crn 4941   Rel wrel 4945   -->wf 5514   -onto->wfo 5516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-br 4393  df-opab 4451  df-xp 4946  df-rel 4947  df-cnv 4948  df-dm 4950  df-rn 4951  df-fun 5520  df-fn 5521  df-f 5522  df-fo 5524
This theorem is referenced by: (None)
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