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 Th19:
  m in Maps(A,B) iff dom m = A & cod m = B
proof
A1: m`2 is Function of dom m,cod m by Th9;
  thus m in Maps(A,B) implies dom m = A & cod m = B
  proof
    assume m in Maps(A,B);
    then
A2: m = [[A,B],m`2] by Th16;
    thus thesis by A2;
  end;
  cod m <> {} or dom m = {} by Th9;
  then [[dom m,cod m],m`2] in Maps(dom m,cod m) by A1,Th15;
  hence thesis by Th8;
end;
