Problem Let $F[x,y,z], F[t]$ be the polynomial ring of 3 unknowns and 1 unknown respectively on a field $F$. Define a ring homomorphism $\phi$: \begin{align*} \phi :F\left[ x,y,z \right] &\rightarrow F\left[ t \right] , \\ f\left( x,y,z \right) &\mapsto f\left( t^2,t^3,t^4 \right) . \end{align*}
Prove that $\ker(\phi)$ is an ideal generated by $y^2-x^3$ and $z-x^2$.
My thought
Denote ideal generated by $y^2-x^3$ and $z-x^2$ by $\left<y^2-x^3,z-x^2\right>:=\{p(y^2-x^3)+q(z-x^2)\mid p,q\in F[x,y,z]\}$. I try to prove that $$\ker(\phi)\supseteq\left<y^2-x^3,z-x^2\right>\cdots\cdots(\#1)$$ and $$\ker(\phi)\subseteq\left<y^2-x^3,z-x^2\right>\cdots\cdots(\#2).$$
For $(\#1)$, when $x=t^2,y=t^3,z=t^4$, we have $y^2-x^3=z-x^2=0$. It is immediate that
$$\ker(\phi)\supseteq\left<y^2-x^3,z-x^2\right>.$$
However, I have no idea how to prove $(\#2)$ despite racking my brains. If it is all about single-variable polynomials, division with remainder may work(dividing $p \in \ker(\phi)$ by $y^2-x^3$, $z-x^2$ and showing the remainder$=0$), but I don't know how to do it in a multi-variable situation.
Any help or discussion would be highly appreciated!