Using arrays with foreign element keys is usually a sign of incorrect design. You need to do separate table with one to many relationship.
But technically it is possible. Example of checking array values without triggers. One reusable function with paramethers and dynamic sql. Tested on PostgreSQL 10.5
create schema if not exists test;
CREATE OR REPLACE FUNCTION test.check_foreign_key_array(data anyarray, ref_schema text, ref_table text, ref_column text)
RETURNS BOOL
RETURNS NULL ON NULL INPUT
LANGUAGE plpgsql
AS
$body$
DECLARE
fake_id text;
sql text default format($$
select id::text
from unnest($1) as x(id)
where id is not null
and id not in (select %3$I
from %1$I.%2$I
where %3$I = any($1))
limit 1;
$$, ref_schema, ref_table, ref_column);
BEGIN
EXECUTE sql USING data INTO fake_id;
IF (fake_id IS NOT NULL) THEN
RAISE NOTICE 'Array element value % does not exist in column %.%.%', fake_id, ref_schema, ref_table, ref_column;
RETURN false;
END IF;
RETURN true;
END
$body$;
drop table if exists test.t1, test.t2;
create table test.t1 (
id integer generated by default as identity primary key
);
create table test.t2 (
id integer generated by default as identity primary key,
t1_ids integer[] not null check (test.check_foreign_key_array(t1_ids, 'test', 't1', 'id'))
);
insert into test.t1 (id) values (default), (default), (default); --ok
insert into test.t2 (id, t1_ids) values (default, array[1,2,3]); --ok
insert into test.t2 (id, t1_ids) values (default, array[1,2,3,555]); --error