In the PDF textbook, "A Friendly Introduction to Mathematical Logic 2nd Edition" by Christopher C. Leary and Lars Kristiansen, on page 54, exercise 6, I am asked to do the following:
Given that $\theta$ is some $\mathcal{L}\text{-formula}$ and $\theta_P$ is the propositional version of $\theta$, prove that :
if $\theta$ is not valid, then $\theta_P$ is not a tautology,
i.e.: prove
$$\not\vDash\theta\to\lnot(\forall v)(\bar v(\theta_P)=T)\\ \text{where $v$ is any truth assignment function, and}\\ \text{$\bar{v}$ is the extension of $v$ to any propositional formula.}$$
For clarity, I will provide some definitions given by the book.
$$v:{Var}_P\to\{T,F\}\\ \text{where ${Var}_P$ is the set of all propositional variables.}$$
$$\bar{v}(\phi)= \begin{cases} v(\phi) & \text{if $\phi$ is a propsitional variable} \\ F & \text{if $\phi=(\lnot\alpha)$ and $\bar{v}(\alpha)=T$} \\ F & \text{if $\phi=(\alpha\lor\beta)$ and $\bar{v}(\alpha)=\bar{v}(\beta)=F$}\\ T & \text{otherwise.} \end{cases} $$
$$\text{$\phi$ is a tautology} \iff (\forall v)(\bar{v}(\phi)=T) \\ \text{where $v$ is any truth assignment function.}$$
Looking at the in the exercise solutions in the back of the book on page 294, the solution goes as follows:
Let $\mathcal{L}$ be a first-order language, let $\phi$ be an $\mathcal{L}\text{-formula}$, and let $A_1, \ldots, A_n$ be the propositional variables that occur in the formula $\phi_P$.
Now, each variable in the list $A_1, \ldots, A_n$ corresponds to a subformula of $\phi$. For $i=1,\ldots,n$, let $\psi_i$ denote the subformula that corresponds to $A_i$. For any $\mathcal{L}\text{-structure } \mathfrak{U}$, we define the valuation $v_\mathfrak{U}$ by $$v_\mathfrak{U}(A_i)= \begin{cases} T & \text{if $\mathfrak{U}\vDash\psi_i$} \\ F & \text{otherwise.} \end{cases}$$ (Claim) $$\mathfrak{U}\not\vDash\phi\iff\bar{v}_\mathfrak{U}(\phi_P)=F\text{.}$$ Note that this claim is equivalent to $$\mathfrak{U}\vDash\phi\iff\bar{v}_\mathfrak{U}(\phi_P)=T\text{.}$$ We will now prove the claim by induction over the complexity of $\phi$.
Case: $\phi$ is an atomic formula. The claim follows straightforwardly from the definition of $v_\mathfrak{U}$.
Case: $\phi$ is of the form $(\forall x)(\alpha)$. The claim follows straightforwardly from the definition of $v_\mathfrak{U}$.
Case: $\phi$ is of the form $(\lnot\alpha)$. We have $$\mathfrak{U}\not\vDash(\lnot\alpha)\iff\mathfrak{U}\vDash\alpha\iff\bar{v}_\mathfrak{U}(\alpha_P)=T\iff\bar{v}_\mathfrak{U}((\lnot\alpha)_p)=F$$ The second equivalence follows by our induction hypothesis. Case: $\phi$ is of the form $(\alpha\lor\beta)$. $$\begin{align} \mathfrak{U}\not\vDash &\iff \mathfrak{U}\not\vDash\alpha\text{ and }\mathfrak{U}\not\vDash\beta \\ &\iff \bar{v}_\mathfrak{U}(\alpha_P)=F\text{ and }\bar{v}_\mathfrak{U}(\beta_P)=F && \text{(ind. hyp.)}\\ &\iff \bar{v}_\mathfrak{U}((\alpha\lor\beta)_P)=F \end{align} $$ This completes the proof of (Claim)
Now, assume that the first-order formula $\theta$ is not valid. By the definition of validity, there exists a structure $\mathfrak{U}$ such that $\mathfrak{U}\not\vDash\theta$. By (Claim) there exists a valuation $v$ such that $\bar{v}(\theta_P)=F$. Thus, by the definition of a tautology, $\theta_P$ is not a tautology. Hence, if $\theta_P$ is a tautology, then $\theta$ is valid.
The problem I'm having with this proof is the definition of $v_\mathfrak{U}$. Previously, the author states on page 52,
Notice that if $\beta_P$ is a tautology, then $\beta$ is valid, but the converse of this statement fails. For example, if $\beta$ is $$[(\forall x)(\theta)\land(\forall x)(\theta\to p)]\to(\forall x)(p),$$ then $\beta$ is valid, but $\beta_P$ would be $[A\land B]\to C$, which is certainly not a tautology.
The author builds $v_\mathfrak{U}$ such that if $\mathfrak{U}\vDash\phi$, then $\bar{v}_\mathfrak{U}(\phi_P)=T$, but just because $\phi$ is valid does not mean it is a tautology (as the author admits above). When I interpret $v_\mathfrak{U}$ within the proof, I see it as building a connection between the truth-hood of of $\phi$ within the $\mathcal{L}\text{-structure } \mathfrak{U}$ and the tautologous-ness of $\phi_P$ in propositional logic. Assuming my understanding is correct, this would imply a contradiction within (Claim) which conflicts with the author's statement I quoted above.
In short, this proof given by the author for exercise 6 is speciously incorrect because I believe the author appears to erroneously define function $v_\mathfrak{U}$. I can't see what I'm doing wrong or what I'm not seeing. Can someone explain this proof to me and debunk my misunderstanding?