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, B being
  Subset-Family of S, P being Subset of S st f.:(f"B) is Cover of P holds B is
  Cover of P
proof
  let T,S be non empty set;
  let f be Function of T,S;
  let B be Subset-Family of S;
  let P be Subset of S;
  assume f.:(f"B) is Cover of P;
  then
A1: P c= union (f.:(f"B)) by SETFAM_1:def 11;
  P c= union B
  proof
    let x be object;
    assume x in P;
    then consider Y being set such that
A2: x in Y and
A3: Y in f.:(f"(B)) by A1,TARSKI:def 4;
    ex Z being set st x in Z & Z in B
    proof
      reconsider Y as Subset of S by A3;
      consider Y1 being Subset of T such that
A4:   Y1 in f"(B) and
A5:   Y = f.:Y1 by A3,Def10;
      consider Y2 being Subset of S such that
A6:   Y2 in B & Y1 = f"(Y2) by A4,Def9;
A7:   f.:(f"Y2) c= Y2 by FUNCT_1:75;
      reconsider Y2 as set;
      take Y2;
      thus thesis by A2,A5,A6,A7;
    end;
    hence thesis by TARSKI:def 4;
  end;
  hence thesis by SETFAM_1:def 11;
end;
