reserve V for non empty set,
  A,B,A9,B9 for Element of V;
reserve f,f9 for Element of Funcs(V);
reserve m,m1,m2,m3,m9 for Element of Maps V;

theorem
  Maps V = union the set of all  Maps(A,B)
proof
  set M = the set of all  Maps(A,B);
  now
    let z be object;
    thus z in Maps V implies z in union M
    proof
      assume z in Maps V;
      then consider
      f being Element of Funcs(V),A,B being Element of V such that
A1:   z = [[A,B],f] &( B = {} implies A = {}) & f is Function of A,B by Th4;
A2:   Maps(A,B) in M;
      z in Maps(A,B) by A1,Th15;
      hence thesis by A2,TARSKI:def 4;
    end;
    assume z in union M;
    then consider C being set such that
A3: z in C and
A4: C in M by TARSKI:def 4;
    consider A,B such that
A5: C = Maps(A,B) by A4;
    ex f being Element of Funcs(V) st z = [[A,B],f] & [[A,B],f] in Maps(V
    ) by A3,A5;
    hence z in Maps(V);
  end;
  hence thesis by TARSKI:2;
end;
