Proving $2^{\aleph_0} = {\aleph_0}!$

Question

How can I show the existence of a injection $\phi:\{x|x \subset \omega \} \rightarrow \{f|f:\omega \rightarrow \omega$ is bijective$\} $?

Answer 1

For $x \subset \omega$ let $x^* =\{n+2 : n \in x \}$. Let $\phi(x)$ be a bijection $ f:\omega \to \omega$ where $f(m)=m$ when $m \in x^*$ and $f(m) \ne m$ when $m \not \in x^*$.