Why is $\displaystyle \sum_{y} P(X=x,Y=y) = P(X=x)$?

Question

Why is $\displaystyle \sum_{y}P(X= x, Y = y) = P(X =x)$

I do not seem to be able to understand this. Its on the 3rd line of the image

Answer 1

Why is $\displaystyle \sum_{y} P(X=x,Y=y) = P(X=x)$?

Proof:

Consider the sets $\{X=x,Y=y\}$. Note that $$\bigcup_{y}\{X=x,Y=y\}=\{X=x\}$$

And $\{X=x,Y=y\}$ are disjoint.

Hence $P(\bigcup_{y}\{X=x,Y=y\})= \sum_{y} P(X=x,Y=y) = P(X=x)$