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
  for f being Function holds pr1(dom f,rng f).:f = dom f
proof
  let f be Function;
  now
    let y be object;
    thus y in dom f implies
     ex x being object st x in dom pr1(dom f,rng f) & x in
    f & y = pr1(dom f,rng f).x
    proof
      assume
A1:   y in dom f;
      take [y,f.y];
A2:   f.y in rng f by A1,FUNCT_1:def 3;
      then [y,f.y] in [:dom f,rng f:] by A1,ZFMISC_1:87;
      hence [y,f.y] in dom pr1(dom f,rng f) by Def4;
      thus [y,f.y] in f by A1,FUNCT_1:def 2;
      thus y = pr1(dom f,rng f).(y,f.y) by A1,A2,Def4
        .= pr1(dom f,rng f).[y,f.y];
    end;
    given x being object such that
A3: x in dom pr1(dom f,rng f) and
    x in f and
A4: y = pr1(dom f,rng f).x;
    consider x1,x2 being object such that
A5: x1 in dom f & x2 in rng f and
A6: x = [x1,x2] by A3,ZFMISC_1:84;
    y = pr1(dom f,rng f).(x1,x2) by A4,A6;
    hence y in dom f by A5,Def4;
  end;
  hence thesis by FUNCT_1:def 6;
end;
