How is a probability conditioned on a $\sigma$-algebra defined? i.e. $\mathbb{P}(X=x\mid\mathcal{G})$

Question

Let a $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space with sub-sigma algebra $\mathcal{G}$. I know that the conditional expectation $\mathbb{E}[X\mid \mathcal{G}]$ is defined as the unique r.v. $Y$ such that $Y$ is integrable, $\mathcal{G}$-measurable and for all $G\in\mathcal{G}$ we have,

$$\int_GXdP=\int_GYdP $$

But how is $\mathbb{P}(X=x\mid \mathcal{G})$ defined? Is this just $\mathbb{E}[1_{X=x}\mid \mathcal{G}]$?