A filter on a poset if by definition its nonempty subset $F$ such that it does not contain the least element and $A, B \in F \Leftrightarrow \exists Z \in F : (Z \le A \wedge Z\le B)$ for every elements $A$, $B$ of our poset.
I will denote the class of all filters (on all posets) as $\mathfrak{F}$.
By definition an ideal is an image of a filter under the inclusion map into dual poset.
I will denote the class of all ideals (on all posets) as $\mathfrak{I}$.
Please help to find a short elegant proof of the following (desirably, using the above definition of ideal instead of the standard definition):
Statement $\mathfrak{J}\cap P = \theta[\mathfrak{F}\cap P]$ for every set $P$ of posets and a self-inverse order reversing isomorphism $\theta$ of $P$.