PostgreSQL two dimensional array intersection

I thought of using functions from the extension intarray, but:

Many of these operations are only sensible for one-dimensional arrays. Although they will accept input arrays of more dimensions, the data is treated as though it were a linear array in storage order.

My other ideas:

Quick and dirty

WITH x(id, arr) AS (VALUES
    (1, '{{1,2}, {3,4}, {4,5}, {4,7}}'::int[])
   ,(2, '{{4,2}, {7,4}, {8,5}, {9,7}}')
   )
SELECT * 
FROM   x
WHERE  arr::text LIKE '%{4,5}%';

The simple trick is to transform the array to its text representation and check with LIKE.

With array-functions

WITH x(id, arr) AS (
    VALUES
     (1, '{{1,2}, {3,4}, {4,5}, {4,7}}'::int[])
    ,(2, '{{4,2}, {7,4}, {8,5}, {9,7}}')
    )
    ,y AS (
    SELECT id, arr, generate_subscripts(arr, 1) AS i
    FROM   x
)
SELECT id, arr
FROM   y
WHERE  arr[i:i] = '{{4,5}}'::int[];

Or, the same in different notation, building on your example:

SELECT id, arr
FROM  (
    SELECT id, arr, generate_subscripts(arr, 1) AS i
    FROM   tbl
    ) x
WHERE  arr[i:i] = '{{4,5}}'::int[];

This generates the array subscripts for the first dimension. With the help of this set-returning function we check each array-slice. Note the double brackets on the right hand side of the expression to create a two-dimensional array that matches.

If {4,5} can be in arr multiple times, add GROUP BY or a DISTINCT clause to avoid duplicate rows.