Random Probability of Random Variable

Let be a Probability Space
Let be a discreet set.
Let be a Random Variable
Let be given by:

i.e. the distribution of .
Consider now the Random Variable given by:

often written as just .
From the definition of , you might naively substitute:

This is wrong.
We need to fully expand the definitions to understand why.
Firstly:

Now we can properly see:

i.e. is randomly sent to a probability in the image of

Thinking about it further,
if takes value with probability
then will take the value with probability