reserve P,Q,X,Y,Z for set, p,x,x9,x1,x2,y,z for object;
reserve D for non empty set;

theorem Th70:
  for f being PartFunc of X,Y st Y = {} implies X = {} ex g being
  Function of X,Y st for x being object st x in dom f holds g.x = f.x
proof
  let f be PartFunc of X,Y such that
A1: Y = {} implies X = {};
  per cases;
  suppose Y = {};
    then reconsider g = f as Function of X,Y by A1;
    take g;
    thus thesis;
  end;
  suppose
A2: Y <> {};
    deffunc F(object) = f.$1;
    defpred P[object] means $1 in dom f;
    set y = the Element of Y;
    deffunc G(object) = y;
A3: for x being object st x in X
holds (P[ x] implies F(x) in Y) & (not P[ x] implies G
    (x) in Y) by A2,PARTFUN1:4;
    consider g being Function of X,Y such that
A4: for x being object st x in X holds (P[ x] implies g.x = F(x)) & (not P[ x]
    implies g.x = G(x)) from Lambda1C(A3);
    take g;
    thus thesis by A4;
  end;
end;
