theorem
  x in f"Y iff [x,f.x] in f & f.x in Y
proof
  thus x in f"Y implies [x,f.x] in f & f.x in Y
  proof
    assume x in f"Y;
    then ex y being object st [x,y] in f & y in Y by RELAT_1:def 14;
    hence thesis by FUNCT_1:1;
  end;
  thus thesis by RELAT_1:def 14;
end;
