reserve m for Cardinal,
  A,B,C for Ordinal,
  x,y,z,X,Y,Z,W for set,
  f for Function;
reserve f,g for Function,
  L for Sequence,
  F for Cardinal-Function;

theorem Th22:
  W is Tarski & (X in W & W is epsilon-transitive or X in W & X c=
  W or card X in card W & X c= W) implies Funcs(X,W) c= W
proof
  assume
A1: W is Tarski;
  assume that
A2: X in W & W is epsilon-transitive or X in W & X c= W or card X in
  card W & X c= W;
A3: card X in card W by A1,A2,Th1;
  let x be object;
  assume
A4: x in Funcs(X,W);
  then consider f such that
A5: x = f and
A6: dom f = X and
A7: rng f c= W by FUNCT_2:def 2;
A8: X c= W by A2;
A9: f c= W
  proof
    let y be object;
    assume
A10: y in f;
    then consider y1,y2 being object such that
A11: [y1,y2] = y by RELAT_1:def 1;
A12: y1 in dom f by A10,A11,FUNCT_1:1;
    y2 = f.y1 by A10,A11,FUNCT_1:1;
    then y2 in rng f by A12,FUNCT_1:def 3;
    hence thesis by A1,A8,A6,A7,A11,A12,Th3;
  end;
  card f = card X by A4,A5,CARD_2:3;
  hence thesis by A1,A3,A5,A9,CLASSES1:1;
end;
