Conditional probability of functions. Small question

Hey, i was just looking at a question where the question is calculate:
P( B(t) =< x| x=B(s) ). That is, calculate the probability that B(t) is less than or equal to x given that x is equal to B(s). Here B(t) is a brownian motion, but my question is not related to this.

I understand the procedure that was given in the solutions, but i skipped some steps by
saying that P( B(t) =< x| x=B(s) ) = P ( B(t) =< B(s) ). So i am just wondering if im allowed to do this step, if B(s) was an arbitrary function?
I just feel that this must be true, but im not sure how to prove or disprove it.