reserve X,X1,X2,Y,Y1,Y2 for set, p,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve f,g,g1,g2,h for Function,
  R,S for Relation;

theorem
  f.:(f"Y) = Y /\ f.:(dom f)
proof
  f.:(f"Y) c= Y & f.:(f"(Y)) c= f.:(dom f) by Th74,RELAT_1:114;
  hence f.:(f"Y) c= Y /\ f.:(dom f) by XBOOLE_1:19;
  let y be object;
  assume
A1: y in Y /\ f.:(dom f);
  then y in f.:(dom f) by XBOOLE_0:def 4;
  then consider x being object such that
A2: x in dom f and
  x in dom f and
A3: y = f.x by Def6;
  y in Y by A1,XBOOLE_0:def 4;
  then x in f"Y by A2,A3,Def7;
  hence thesis by A2,A3,Def6;
end;
