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 T, P being Subset of T st B is Cover of P holds f"(f.: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 T;
  let P be Subset of T;
  assume B is Cover of P;
  then
A1: P c= union B by SETFAM_1:def 11;
  P c= union(f"(f.: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 B by A1,TARSKI:def 4;
    ex Z being set st x in Z & Z in f"(f.:B)
    proof
      reconsider Y as Subset of T by A3;
      set Z = f"(f.:Y);
      take Z;
      dom f = T by Def1;
      then
A4:   Y c= f"(f.:Y) by FUNCT_1:76;
      f.:Y in f.:B by A3,Def10;
      hence thesis by A2,A4,Def9;
    end;
    hence thesis by TARSKI:def 4;
  end;
  hence thesis by SETFAM_1:def 11;
end;
