Metamath Proof Explorer

Theorem foconst

Description: A nonzero constant function is onto. (Contributed by NM, 12-Jan-2007)

Ref Expression
Assertion foconst 
|- ( ( F : A --> { B } /\ F =/= (/) ) -> F : A -onto-> { B } )

Proof

Step Hyp Ref Expression
1 frel 
 |-  ( F : A --> { B } -> Rel F )
2 relrn0 
 |-  ( Rel F -> ( F = (/) <-> ran F = (/) ) )
3 2 necon3abid 
 |-  ( Rel F -> ( F =/= (/) <-> -. ran F = (/) ) )
4 1 3 syl 
 |-  ( F : A --> { B } -> ( F =/= (/) <-> -. ran F = (/) ) )
5 frn 
 |-  ( F : A --> { B } -> ran F C_ { B } )
6 sssn 
 |-  ( ran F C_ { B } <-> ( ran F = (/) \/ ran F = { B } ) )
7 5 6 sylib 
 |-  ( F : A --> { B } -> ( ran F = (/) \/ ran F = { B } ) )
8 7 ord 
 |-  ( F : A --> { B } -> ( -. ran F = (/) -> ran F = { B } ) )
9 4 8 sylbid 
 |-  ( F : A --> { B } -> ( F =/= (/) -> ran F = { B } ) )
10 9 imdistani 
 |-  ( ( F : A --> { B } /\ F =/= (/) ) -> ( F : A --> { B } /\ ran F = { B } ) )
11 dffo2 
 |-  ( F : A -onto-> { B } <-> ( F : A --> { B } /\ ran F = { B } ) )
12 10 11 sylibr 
 |-  ( ( F : A --> { B } /\ F =/= (/) ) -> F : A -onto-> { B } )