unique amount more than one table

I have five tables each with a "name" field. Due to limitations in my user interface, I want a name to be unique amoung these five tables.

I thought I could first create a view with something like:

SELECT name, 'table1' as type from table1
UNION ALL
SELECT name, 'table2' as type from table2
UNION ALL
SELECT name, 'table3' as type from table3
...

I called this view xxx (I'm just experimenting right now).

I then created a function:

CREATE OR REPLACE FUNCTION unique_xxx ( ) RETURNS boolean AS $$
SELECT ( SELECT max(cnt) FROM ( SELECT count(*) AS cnt FROM xxx GROUP BY name ) AS foo ) = 1;
$$ LANGUAGE SQL;

Next I added a check constraint with:

ALTER TABLE table1 ADD CHECK ( unique_xxx() );

A test shows:

select unique_xxx();
unique_xxx
------------
t
(1 row)

After I insert a row that I want to be rejected, I can do:

select unique_xxx();
unique_xxx
------------
f
(1 row)

but the insert was not rejected. I'm guessing because the check constraint runs before the insert? So, I could change my approach and have my unique_xxx function see if the name to be added is already in the xxx view but it is at that point that I stopped and thought I would ask for advice. Am I close or am I going down the wrong road?

Thank you for your time,
pedz