reserve f,g,h for Function,
  A for set;
reserve F for Function,
  B,x,y,y1,y2,z for set;
reserve x,z for object;

theorem Th11:
  (for z st z in dom f holds f.z = x) implies f = dom f --> x
proof
  assume
A1: for z st z in dom f holds f.z = x;
  now
    per cases;
    suppose
A2:   dom f = {};
      thus thesis by A2;
    end;
    suppose
A3:   dom f <> {};
      set z = the Element of dom f;
      now
        let y be object;
        thus y in {x} implies ex y1 being object st y1 in dom f & y = f.y1
        proof
          assume y in {x};
          then y = x by TARSKI:def 1;
          then f.z = y by A1,A3;
          hence thesis by A3;
        end;
        assume ex y1 being object st y1 in dom f & y = f.y1;
        then y = x by A1;
        hence y in {x} by TARSKI:def 1;
      end;
      then rng f = {x} by FUNCT_1:def 3;
      hence thesis by Th9;
    end;
  end;
  hence thesis;
end;
