reserve A, B, X, Y for set;

theorem
  (delta X).:A c= [:A,A:]
proof
  set f = delta X;
  let y be object;
  assume y in f.:A;
  then consider x being object such that
A1: x in dom f and
A2: x in A and
A3: y = f.x by FUNCT_1:def 6;
  y = [x,x] by A1,A3,FUNCT_3:def 6;
  hence thesis by A2,ZFMISC_1:def 2;
end;
