# Rice's Theorem for Turing machine with fixed output

So I was supposed to prove with the help of Rice's Theorem whether the language: $$L_{5} = \{w \in \{0,1\}^{*}|\forall x \in \{0,1\}^{*}, M_{w}(w) =x\}$$ is decidable.

First of all: I don't understand, why we can use Rice's Theorem in the first place: If I would chose two Turingmachines $$M_{w}$$ and $$M_{w'}$$ with $$w \neq w'$$ then I would get $$M_{w}(w) = M_{w'}(w) = x$$. But this would lead to $$w'$$ not being in $$L_{5}$$ and $$w \in L_{5}$$. Or am I misunderstanding something?

Second: The solution says, that the Language $$L_{5}$$ is decidable as $$L_{5} = \emptyset$$ because the output is clearly determined with a fixed input. But why is that so? I thought that $$L_{5}$$ is not empty because there are TM which output x on their own input and there are some which do not.

A word $$w$$ belongs to $$L_5$$ if for all $$x \in \{0,1\}^*$$ it is the case that $$M_w(w) = x$$. In particular, if $$w \in L_5$$ then $$M_w(w) = 0$$ and $$M_w(w) = 1$$, which can't both be true. So no word belongs to $$L_5$$.