Confusion about $\mathbb E[Y\mid X]$.

Question

Let $\varphi :\mathbb R\to \mathbb R$ defined by $$\varphi (x)=\mathbb E[Y\mid X=x].$$ Then, $\mathbb E[Y\mid X]$ is defined by $\varphi (X)$. This definition looks a bit confusion, rigorously speaking, shouldn't it be $$\varphi (X)=\mathbb E[Y\mid X=X],$$ and thus $$\mathbb E[Y\mid X=X](\omega ):=\varphi (X)(\omega )=\varphi (X(\omega ))=\mathbb E[Y\mid X=X(\omega )] \ \ ?$$