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 Th25:
  for f being Function holds union(("f).:B) c= f"(union B)
proof
  let f be Function;
  let x be object;
  assume x in union(("f).:B);
  then consider X such that
A1: x in X and
A2: X in ("f).:B by TARSKI:def 4;
  consider Y being object such that
A3: Y in dom("f) and
A4: Y in B and
A5: X = "f.Y by A2,FUNCT_1:def 6;
  reconsider Y as set by TARSKI:1;
A6: ("f).Y = f"Y by A3,Th21;
  Y in bool rng f by A3,Def2;
  then
A7: f.:X in B by A4,A5,A6,FUNCT_1:77;
A8: f"Y c= dom f by RELAT_1:132;
  then f.x in f.:X by A1,A5,A6,FUNCT_1:def 6;
  then f.x in union B by A7,TARSKI:def 4;
  hence thesis by A1,A5,A6,A8,FUNCT_1:def 7;
end;
