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 B c= bool rng f holds union(("f).:B) = f"( union B )
proof
  let f be Function such that
A1: B c= bool rng f;
A2: f"(union B) c= union(("f).:B)
  proof
    let x be object;
    assume
A3: x in f"(union B);
    then f.x in union B by FUNCT_1:def 7;
    then consider Y such that
A4: f.x in Y and
A5: Y in B by TARSKI:def 4;
    x in dom f by A3,FUNCT_1:def 7;
    then
A6: x in f"Y by A4,FUNCT_1:def 7;
    Y in bool rng f by A1,A5;
    then
A7: Y in dom("f) by Def2;
    ("f).Y = f"Y by A1,A5,Def2;
    then f"Y in ("f).:B by A5,A7,FUNCT_1:def 6;
    hence thesis by A6,TARSKI:def 4;
  end;
  union(("f).:B) c= f"(union B) by Th25;
  hence thesis by A2,XBOOLE_0:def 10;
end;
