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

theorem
  for f being Function of X,Y for y st y in f.:P ex c being Element of X
  st c in P & y = f.c
proof
  let f be Function of X,Y;
  let y;
  assume y in f.:P;
  then consider x such that
A1: x in X and
A2: x in P & y = f.x by Th63;
  reconsider c = x as Element of X by A1;
  take c;
  thus thesis by A2;
end;
