What is perfect secrecy?
The Lindell and Katz, in their book, give the definition as follows;


  An encryption scheme $(Gen,Enc,Dec)$ over a message space
  $\mathcal{M}$ is perfectly secret if for every probability
  distribution of over $\mathcal{M}$, every every message
  $m\in\mathcal{M}$, and every ciphertext $c \in \mathcal{C}$ for which
  $Pr[C=c]&gt;0$;
  
  $$Pr[M=m|C=c] = Pr[M=m]$$


We can view this definition as the distribution over messages and ciphertext are independent.



The proof used this definition and the equivalent definition of perfect secrecy

$Pr[C=c|M=m] = Pr[C=c]$

and, this can be said, the ciphertext reveals no information about the plaintext other than the size.



On your proof;

On the third P, on the numerator and denominator; 

$Pr[Enc_k(m_1)=c] = Pr[Enc_k(M) =c | M=m_1]$