Why can't different messages result in the same ciphertext when encrypted with RSA? In the encryption step of the RSA encryption to get the cypher text $C$

$C \equiv M^e \pmod n$

I am struggling to understand how we know that there only exists one $M$ that maps to a certain $C$?

Does there not exist an $M$ and an $N$ such that $M^e \equiv N^e \pmod n$? How do we know this?
The property that you are talking about is the injectivity. I think this link will maybe help you.