No I dont think I understand the second step in your proof.

You say f(x) - x = 0

Then you say f(f(x)) - x = 0 ???

How

The idea is functions obey "axiom of substitution",(AOS)

that is if

So if f(x) = x for some x,

Then by AOS, f(f(x)) = f(x) = x for those x