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

theorem Th63:
  for P being set for f being Function of X,Y
   for y holds y in f.:P implies ex x st x in X & x in P & y = f.x
proof
  let P be set;
  let f be Function of X,Y;
  let y;
  assume y in f.:P;
  then consider x being object such that
A1: x in dom f and
A2: x in P & y = f.x by FUNCT_1:def 6;
   reconsider x as set by TARSKI:1;
  take x;
  thus x in X by A1;
  thus thesis by A2;
end;
