let L be non empty up-complete Poset; :: thesis: for f being Function of L,L st f is idempotent & f is directed-sups-preserving holds
Image f is directed-sups-inheriting
let f be Function of L,L; :: thesis: ( f is idempotent & f is directed-sups-preserving implies Image f is directed-sups-inheriting )
set S = subrelstr (rng f);
set a = the Element of L;
reconsider S9 = subrelstr (rng f) as non empty full SubRelStr of L ;
assume A1: ( f is idempotent & f is directed-sups-preserving ) ; :: thesis: Image f is directed-sups-inheriting
subrelstr (rng f) is directed-sups-inheriting

proof

let X be directed Subset of (subrelstr (rng f)); :: according to WAYBEL_0:def 4 :: thesis: ( X = {} or not ex_sup_of X,L or "\/" (X,L) in the carrier of (subrelstr (rng f)) )
X c= the carrier of (subrelstr (rng f)) ;
then A2: X c= rng f by YELLOW_0:def 15;
assume X <> {} ; :: thesis: ( not ex_sup_of X,L or "\/" (X,L) in the carrier of (subrelstr (rng f)) )
then X is non empty directed Subset of S9 ;
then reconsider X9 = X as non empty directed Subset of L by YELLOW_2:7;
assume A3: ex_sup_of X,L ; :: thesis: "\/" (X,L) in the carrier of (subrelstr (rng f))
f preserves_sup_of X9 by A1;
then sup (f .: X9) = f . (sup X9) by A3;
then sup X9 = f . (sup X9) by A1, A2, YELLOW_2:20;
then "\/" (X,L) in rng f by FUNCT_2:4;
hence "\/" (X,L) in the carrier of (subrelstr (rng f)) by YELLOW_0:def 15; :: thesis: verum

end;

hence Image f is directed-sups-inheriting ; :: thesis: verum