If $F$ is valid then $F \cup \{res(C_1,C_2,A_i)\}$ is valid I have to prove the following problem in propositional logic:

Let $F$ be a set of clauses and let $F' = F \cup \{res(C_1,C_2,A_i)\}$ be the extension of $F$ by a resolvent of some clauses $C_1,C_2 \in F$ where $A_i$ is a literal occuring positively in $C_1$ and negatively in $C_2$.

Prove that: If $F$ is valid, then $F'$ is valid.

So in other words I have to prove that when I construct the union of the original formula $F$ and the formula resulting by applying resolution on $F$ over a literal $A_i$ in $F$, validity is still preserved.

I think that this should be provable by applying a direct proof.

Recall: Note that resolution is defined as follows: given two clauses:
$C_1 = (A_1 \lor \dots \lor A_i \lor \dots \lor A_n)$ and $C_2 = (B_1 \lor \dots \lor B_j \lor \dots \lor B_m)$ such that for some $i, j$ with $1 \leq i \leq n$, and $1 \leq j \leq m,\; A_i = \neg B_j$,
the resolvent of $C_1$ and $C_2$ on $A_i$ is the clause

$res(C_1,C_2,A_i) = (A_1 \lor \dots \lor A_{i-1} \lor A_{i+1} \lor \dots \lor A_n \lor B_1 \lor \dots B_{j-1} \lor B_{j+1} \lor \dots \lor B_m)$

EDIT: Here is my try:

Let $I$ be an interpratation taken from the set of models $Mod(F)$. Hence, $I(F) = 1$. Because this interpretation satisfies $F$ it must also be the case that it satisfies $C_1$ and $C_2$. If we take a look at the structure of $C_1$ and $C_2$ we have to distinguish between $2$ different cases:

(1) $A_i$ is positive in $C_1$ but negative in $C_2$ and

(2) $A_i$ is negative in $C_1$ but positive in $C_2$.

In case (1) if $A_i$ is set to true in $C_1$ it would be false in $C_2$.
In case (2) if $A_i$ is set to true in $C_2$ it would be false in $C_1$.

$A_i$ cannot be the literal which preserves satisfiability of $F$. Hence $I$ must include some assignment of other literals in $F$ such that $F$ is satisfied. Thus, the resolvant $res(C_1,C_2,A_i)$ is also satisfied.
Since you are only asking for a hint:

You have to prove that, for any interpretation I, if it satisfies all
the clauses in $F$, then it also satisfies the resolvant
$res(C_1,C_2,A_i)$, and hence it satisfies all the clauses in $F'$.

From that you can infer the result to be proved, with some universal quantification.

Note on terminology and notations:


I would rather say that, when $F$ is valid, every interpretation
satisfies it. It is $F$ that is to be satisfied, not the interpretation. Or more generally, it may also be a clause $C$, a literal $A_i$, or a set of clauses $F$ that may (have to) be satisfied.
$F$ is valid iff for any interpretation $I$, $I(F)=1$.
You should not say "for any interpretation $I(F)$", since as you noted yourself, $I(F)$ is a truth value. An interpretation in propositional logic is (usually) an assignment of truth values to the symbols used in the clauses, i.e. a mapping of the set of symbols to the set of truth values. It also implies interpreting logical connectives in the usual way, and sets of clauses as a conjunction of the clauses. Then, truth tables allow you to associate truth values to each construct, up to the whole set of clauses, which is what is meant by $I(F)$. The interpretation is $I$. while $I(F)$ is the truth value it gives to the set $F$ of clauses.


Interpretation may be defined in a more general way, but the above should be enough here.