Let R and S be two rings with unit elements. Let us assume that the unit element of R is $1\_R$
and the unit element of S is $1\_S$. Let $\phi$ : R → S be a ring homomorphism. Show that if S is
an integral domain, then $\phi$ ($1\_R$) = $1\_S$.

Note: $\phi$($1\_R$) $\neq 0\_S$ (Zero element in S)

I want to prove this **via contradiction specifically** and have proceeded as follows:

$\exists y \in S \ \ s.t.$ $ y \ \phi(1\_R) \neq \ y $

Then-

$ y \ \phi(1\_R)- y \neq 0\_S $

I am stuck here and don't know how to proceed ahead with this proof using contradiction.