understanding over the proof of a theorem

Question

Let $K$ be a field and let $P\subset K$ its prime field (or minimal field).Then:

1. $char K=p>0 \iff P\simeq\Bbb F_p$ where p is a prime number
2. $char K=0 \iff P\simeq\Bbb Q$

Therefore , barring isomorphisms, the only prime fields are $\Bbb Q$ and the fields $\Bbb F_p$

For the proof we consider the canonical homomorphism of rings $\varphi:\Bbb Z\rightarrow K$:it factors through the prime field $P\subset K$ thus hold $im \varphi\subset P$ .

The reverse implication is trivial($P\subset im\varphi$); but my question is why we can assert that $im\varphi \subset \Bbb P$ ?