reserve p,q,x,x1,x2,y,y1,y2,z,z1,z2 for set;
reserve A,B,V,X,X1,X2,Y,Y1,Y2,Z for set;
reserve C,C1,C2,D,D1,D2 for non empty set;

theorem Th9:
  for f being Function holds rng(.:f) c= bool rng f
proof
  let f be Function;
  let y be object;
   reconsider yy=y as set by TARSKI:1;
  assume y in rng(.:f);
  then consider x being object such that
A1: x in dom(.:f) & y = (.:f).x by FUNCT_1:def 3;
   reconsider x as set by TARSKI:1;
  y = f.:x by A1,Th7;
  then yy c= rng f by RELAT_1:111;
  hence thesis;
end;
