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 Th27:
  for f being Function holds union(("f)"A) c= f.:(union A)
proof
  let f be Function;
  let y be object;
  assume y in union(("f)"A);
  then consider Y such that
A1: y in Y and
A2: Y in ("f)"A by TARSKI:def 4;
  dom("f) = bool rng f by Def2;
  then
A3: Y in bool rng f by A2,FUNCT_1:def 7;
  then consider x being object such that
A4: x in dom f & y = f.x by A1,FUNCT_1:def 3;
  "f.Y in A by A2,FUNCT_1:def 7;
  then
A5: f"Y in A by A3,Def2;
  x in f"Y by A1,A4,FUNCT_1:def 7;
  then x in union A by A5,TARSKI:def 4;
  hence thesis by A4,FUNCT_1:def 6;
end;
