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
  for f being Function st A c= bool dom f & f is one-to-one holds union(
  ("f)"A) = f.:(union A)
proof
  let f be Function such that
A1: A c= bool dom f and
A2: f is one-to-one;
A3: f.:(union A) c= union(("f)"A)
  proof
    let y be object;
    assume y in f.:(union A);
    then consider x being object such that
A4: x in dom f and
A5: x in union A and
A6: y = f.x by FUNCT_1:def 6;
    consider X such that
A7: x in X and
A8: X in A by A5,TARSKI:def 4;
A9: f"(f.:X) c= X by A2,FUNCT_1:82;
A10: f.:X c= rng f by RELAT_1:111;
    then f.:X in bool rng f;
    then
A11: (f.:X) in dom("f) by Def2;
    X c= f"(f.:X) by A1,A8,FUNCT_1:76;
    then f"(f.:X) = X by A9,XBOOLE_0:def 10;
    then ("f).(f.:X) in A by A8,A10,Def2;
    then
A12: (f.:X) in ("f)"A by A11,FUNCT_1:def 7;
    y in (f.:X) by A4,A6,A7,FUNCT_1:def 6;
    hence thesis by A12,TARSKI:def 4;
  end;
  union(("f)"A) c= f.:(union A) by Th27;
  hence thesis by A3,XBOOLE_0:def 10;
end;
