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Theorem foconst 5478
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 5408 . . . . 5  |-  ( F : A --> { B }  ->  Rel  F )
2 relrn0 4953 . . . . . 6  |-  ( Rel 
F  ->  ( F  =  (/)  <->  ran  F  =  (/) ) )
32necon3abid 2492 . . . . 5  |-  ( Rel 
F  ->  ( F  =/=  (/)  <->  -.  ran  F  =  (/) ) )
41, 3syl 15 . . . 4  |-  ( F : A --> { B }  ->  ( F  =/=  (/) 
<->  -.  ran  F  =  (/) ) )
5 frn 5411 . . . . . 6  |-  ( F : A --> { B }  ->  ran  F  C_  { B } )
6 sssn 3788 . . . . . 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 5471 . 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 1632    =/= wne 2459    C_ wss 3165   (/)c0 3468   {csn 3653   ran crn 4706   Rel wrel 4710   -->wf 5267   -onto->wfo 5269
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277
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