Improving a filter array function with Set and objects, to match elements and conditions
The Array methods filter() and every() come in very handy here. First you could define a function to check wheter a given element matches all conditions:
function meetsAll(element, conditions) {
    return conditions.every(c =&gt; element.condition_matrix.includes(c));
}

Now it's almost trivial to filter out the matching elements using Array.filter():
const meetConditions = this.arrayWithAllElements.filter(
    element =&gt; meetsAll(element, userSelectedConditions)
);

I recommend you also have a look at the other functional methods defined on Array.