$$X|Y=y$$ is the random variable X when conditioned on the realization of Y=y.

$$1(Y=y)$$ is the indicator function whether the random variable Y equals y.

My intuition very strongly agrees, whether Y is y or not should have nothing to do with what I draw from X when I condition on Y=y. However I am not able to formalize it.

Edit: as an example, whether it rained yesterday has nothing to with the chance of a rain today GIVEN that it rained yesterday.