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 Th16:
  for f being Function holds union((.:f)"B) c= f"(union B)
proof
  let f be Function;
  let x be object;
  assume x in union((.:f)"B);
  then consider X such that
A1: x in X and
A2: X in (.:f)"B by TARSKI:def 4;
  dom(.:f) = bool dom f by Def1;
  then
A3: X in bool dom f by A2,FUNCT_1:def 7;
  then
A4: f.x in f.:X by A1,FUNCT_1:def 6;
  (.:f).X in B by A2,FUNCT_1:def 7;
  then f.:X in B by A3,Def1;
  then f.x in union B by A4,TARSKI:def 4;
  hence thesis by A1,A3,FUNCT_1:def 7;
end;
