reserve P,Q,X,Y,Z for set, p,x,x9,x1,x2,y,z for object;
reserve D for non empty set;
reserve A,B for non empty set;

theorem
  for T,S being non empty set, f being Function of T,S, Q being
  Subset-Family of S holds union(f.:(f"(Q))) c= union Q
proof
  let T,S be non empty set;
  let f be Function of T,S;
  let Q be Subset-Family of S;
  let x be object;
  thus x in union(f.:(f"(Q))) implies x in union Q
  proof
    assume x in union(f.:(f"Q));
    then consider A being set such that
A1: x in A and
A2: A in f.:(f"Q) by TARSKI:def 4;
    reconsider A as Subset of S by A2;
    consider A1 being Subset of T such that
A3: A1 in f"Q and
A4: A = f.:A1 by A2,Def10;
    consider A2 being Subset of S such that
A5: A2 in Q & A1 = f"A2 by A3,Def9;
    f.:(f"A2) c= A2 by FUNCT_1:75;
    hence thesis by A1,A4,A5,TARSKI:def 4;
  end;
end;
