[...]
> I need to search a table to find sets of rows that have a column matching
> itself for the whole set and another column matching row for row with a
> list I am going to supply. The result I should receive should be value of
> the column that matches itself.
[...]

How about:

DROP TABLE IF EXISTS T;

CREATE TABLE T
(
    id  int,
    val int
);

INSERT INTO T (id, val) VALUES
(3,  1),
(3,  2),
(4,  8),
(4,  9),
(4,  10);

SELECT
    id
FROM
    T
GROUP BY
    id
HAVING
    array_agg(val) = ARRAY[8, 9, 10];


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