Proposition $4.8$. Let $(D, ≤)$ be a directed set and $(X, Φ)$ be a uniform space. Let $I, K$ be $D$-admissible ideals such that $K ⊆ I$.Suppose that every $I^K$-Cauchy net $x: D → X$ is $I^K$-convergent. Then also every Cauchy net $x: D → X$ on the directed set $D$ is convergent.
Proof:
Let $x: D → X$ be a Cauchy net. Since $I_D ⊆ K$, it is also $K$-Cauchy (Lemma 3.3) and, consequently, it is $I^K$-Cauchy (Lemma $3.6$).According to our assumptions, the net $x$ is then also $I^K$-convergent to some point $l \in X$. This means that there is a set $M ∈ F(I)$ such that $x|_M$ is $K|_M$-convergent to $l$.That is, for every neighborhood $U[l]$ of $l$ we have $$x^{−1}(U[l])=F∩M$$ for some $F ∈ F(K)$. Since $F(K) ⊆ F(I)$, we get that $F ∩ M ∈ F(I)$ and consequently, $F ∩ M$ is cofinal in $D$. Now the claim follows from Lemma $4.3$.
Now my confusion is the equation $$\color{blue}{x^{−1}(U[l])=F∩M}$$ because I think it should be an inequation $$\color{blue}{x^{−1}(U[l])\supset F∩M}\\ \color{blue}{\text{or}}\\ \color{blue}{x^{−1}(U[l])\cap M=F∩M}$$
Here is why:
$x$ is $I^K$-convergent to $l\in X$ $\implies$ the sequence $y:D\rightarrow X$ given by $$y_{_d} = \begin{cases} x_{_d}, & \text{if $d\in M$} \\ l, & \text{if $d\notin M$} \end{cases};\text{ for some }M\in F(I)$$ is $K$-convergent to $l.$ Which means for any neighbourhood $U[l]$ of $l$ , the set $$\{d\in D:y_{_d}\notin U[l]\}\in K \\ \implies \{d\in D: y_{_d}\in U[l]\}\in F(K)\subset F(I).$$ Also $$\{d\in D: y_{_d}\in U[l]\}=\{d\in M: x_{_d}\in U[l]\}\cup (D\backslash M)=(x^{-1}(U[l])\cap M)\cup (D\backslash M)...........(*)$$
Then $$x^{-1}(U[l])=\{d\in D: (x_{_d},l)\in U\}\\=\{d\in D: x_{_d}\in U[l]\}\\=(x^{-1}(U[l]\cap M))\cup (x^{-1}(U[l]\cap(D\backslash M)))...........(**)$$
Now chances are $(**)\subset (*)$ and $(**)\neq (*)$.
And if we name the set in $(*)$ as $F$ then $F\in F(K)\subset F(I)$ and $F\cap M=x^{-1}(U[l])\cap M$ i.e $x^{-1}(U[l])\supset F\cap M\implies x^{-1}(U[l])\in F(I).$ Rest of the proof is smoothly following , it's just that particular equation I have doubts about.
Here is a link to that paper $I^K$-CAUCHY FUNCTIONS by PRATULANANDA DAS, MARTIN SLEZIAK, AND VLADIM´IR TOMA