MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  foconst Structured version   Unicode version

Theorem foconst 5714
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 5642 . . . . 5  |-  ( F : A --> { B }  ->  Rel  F )
2 relrn0 5173 . . . . . 6  |-  ( Rel 
F  ->  ( F  =  (/)  <->  ran  F  =  (/) ) )
32necon3abid 2628 . . . . 5  |-  ( Rel 
F  ->  ( F  =/=  (/)  <->  -.  ran  F  =  (/) ) )
41, 3syl 16 . . . 4  |-  ( F : A --> { B }  ->  ( F  =/=  (/) 
<->  -.  ran  F  =  (/) ) )
5 frn 5645 . . . . . 6  |-  ( F : A --> { B }  ->  ran  F  C_  { B } )
6 sssn 4102 . . . . . 6  |-  ( ran 
F  C_  { B } 
<->  ( ran  F  =  (/)  \/  ran  F  =  { B } ) )
75, 6sylib 196 . . . . 5  |-  ( F : A --> { B }  ->  ( ran  F  =  (/)  \/  ran  F  =  { B } ) )
87ord 375 . . . 4  |-  ( F : A --> { B }  ->  ( -.  ran  F  =  (/)  ->  ran  F  =  { B } ) )
94, 8sylbid 215 . . 3  |-  ( F : A --> { B }  ->  ( F  =/=  (/)  ->  ran  F  =  { B } ) )
109imdistani 688 . 2  |-  ( ( F : A --> { B }  /\  F  =/=  (/) )  -> 
( F : A --> { B }  /\  ran  F  =  { B }
) )
11 dffo2 5707 . 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 366    /\ wa 367    = wceq 1399    =/= wne 2577    C_ wss 3389   (/)c0 3711   {csn 3944   ran crn 4914   Rel wrel 4918   -->wf 5492   -onto->wfo 5494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-opab 4426  df-xp 4919  df-rel 4920  df-cnv 4921  df-dm 4923  df-rn 4924  df-fun 5498  df-fn 5499  df-f 5500  df-fo 5502
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator