No set to which all functions belong
I'm having trouble proving that there is no set of all functions. I tried the approach similar to the approach used to show that there is no set of all sets, but was not able to make this work.
First I assumed the set $A$ of all functions, and then constructed the subset $B$ where each member of the subset is not a member of itself. I could then prove that the subset was not a member of $A$. To complete the proof I need to show that $B$ is a function, but $B$ doesn't seem to be a function.
Would like some guidance on the remaining steps, or a better approach to take.