Like David said, the proof is almost identical to the earlier one for $R$-matrices:
$r(x\otimes 1) = r\circ (id\otimes\mu)(x\otimes 1\otimes 1) = (r_{13}\ast r_{12})(x\otimes 1\otimes1)= \sum r(x'\otimes 1)r(x''\otimes 1) = (r \ast r)(x\otimes 1).$
Since $r$ is invertible, $r(x\otimes 1)=\epsilon(x)$.