Coordinate ring of an affine variety is independent of the embedding in the ambient affine $n$-space This is concerning a remark in page $664$ of Abstract algebra, 3rd edition, by Dummit and Foote. Could someone help me understand the remark  to the effect that the following two statements imply  that the isomorphism type of the coordinate ring of an affine algebraic set $V$ (as a $k$-algebra) does not depend on the embedding of $V$ in a particular affine $n$-space. Here $k$ is a field.
Statement 1 : The morphism $\phi: V \to W$ is an isomorphism if and only if the induced map $\widetilde{\phi}: k[W] \to k[V]$ is an isomorphism of $k$-algebras.
Statement 2 : Suppose $\phi: V \to W$ be a map of affine algebraic sets. Then $\phi$ is a morphism if and only if for every $f: W \to k$ in $k[W]$, the composite map $f \circ \phi$ is an element of $k[V]$ (as a $k$-valued function on $V$).
For example, the "embedding" $\mathbb{A}^1 \to \mathbb{A}^2$ mapping $t$ to $(t, t^2)$ is an embedding of the affine variety $\mathbb{A}^1$ as a parabola in $\mathbb{A}^2$. Here $\mathbb{A}^n$ denotes the affine $n$-space over the field $k$. Clearly, coordinate rings of $\mathbb{A}^1$ which is $k[X]$ and the parabola which is $k[X,Y]/(Y - X^2)$ are isomorphic as $k$-algebras under the explict morphism $t \mapsto (t, t^2) \to t: \mathbb{A}^1 \to \mathbb{A}^2 \to \mathbb{A}^1$. But, I am not entirely sure how to prove in general using the above two statements.
 A: Dummit and Foote are setting up one of the most important trivial facts in 20th century mathematics.
We want to be able to talk about the geometry of a variety without mentioning any particular embedding into affine space. After all, if we define something for a variety $V \subseteq \mathbb{A}^n$, we have to check by hand that it really is a property of $V$, rather than some artifact of how we chose to embed it into space. For instance, the dimension of $V$ is intrinsic, but the codimension depends heavily on the choice of embedding!
It gets really annoying to have to check that every definition we make only depends on $V$ up to isomorphism, so it would be super convenient to have a language that doesn't see embeddings at all. In that case, anything we say will automatically be intrinsic to $V$!
But how should we do that? The key idea, it turns out, is to focus not on the geometric objects $V$ and $W$, but rather on the algebraic objects $k[V]$ and $k[W]$.
First, from theorem 6 (your statement 1), we see that the coordinate ring of a variety doesn't depend on the embedding. What does this mean? Well if $V_1 \subseteq \mathbb{A}^n$ and $V_2 \subseteq \mathbb{A}^m$ are isomorphic, we think of them as being "the same variety", just embedded into space in two different ways. Statement 1 literally says that $V_1 \cong V_2$ as varieties if and only if $k[V_1] \cong k[V_2]$ as $k$-algebras. That is, the $k$-algebra of coordinate functions is the same no matter which embedding you choose ($V_1$, or $V_2$, or any other isomorphic embedding), since any two embeddings give you the same ring.
Of course, there's more to life than regular functions! We want to be able to talk about morphisms of varieties $V \to W$. Corollary 7 (your statement 2) tells us that we can do that as well! A morphism of varieties $V \to W$ is the same thing as a morphism of $k$-algebras $k[W] \to k[V]$. Notice the map $f \mapsto f \circ \phi$ is an algebra homomorphism. Moreover, one can show that every algebra homomorphism is of this form!
But this tells us that if we're interested in varieties up to isomorphism, and morphisms between these varieties, we have a super concrete way of getting our hands on them! Simply forget about the variety $V$ and work directly with the (finitely generated, reduced) $k$-algebra that represents its coordinate ring.
Indeed, the category of $k$-varieties is equivalent to the (opposite of the) category of finitely generated, reduced $k$-algebras! So as long as your question can be phrased "categorically", anything you want to know about varieties can be answered by consulting the coordinate rings directly. This has the benefit of being totally agnostic as the the embedding into affine space, so everything we say is automatically intrinsic. It has the detriment of being more abstract, since now we're doing everything in terms of commutative algebra. It's easy to get lost in the sauce and forget that we're actually doing geometry!
As an aside, if you've ever wondered why people try so hard to phrase every little thing categorically, using language that might seem terribly abstract for the simple thing they want to describe, this is why! We want to be able to say as much about our varieties as we can, but since we can only probe varieties indirectly by asking categorical questions of the $k$-algebras, we want to be able to say as much in the categorical language as possible.

I hope this helps ^_^
