I'm trying to understand the working of RSA algorithm. I am getting confused in the decryption part. I'm assuming
$$n = pq$$ $$m = \phi(n) = (p - 1)(q - 1)$$
E is the encryption key $\gcd(\phi(n), E) = 1$
D is the decryption key, and $DE = 1 \mod \phi(n)$
$x$ is the plain text
Encryption works as ($y = x^E \mod n$) and decryption works as ($x = y^D \mod n$)
The explanation for why the decryption works is that since $DE = 1 + k\phi(n)$,
$$y^D = x^{ED} = x^{1 + k \phi(n)} = x(x^{\phi(n)})^k = x \mod n$$
The reason why last expression works is $x^{\phi(n)} = 1 \mod n$ ? According to Eulers theorem this is true only if $x \text{ and }\phi(n)$ are coprimes. But $x$ is only restricted to be $0 < x < n$ and $\phi(n) < n$. So $x$ should be chosen to be coprime with $\phi(n)$?
Help me clear out the confusion!