Sqlite update with min if smaller than existing

Must misunderstand something basic here, have a couple tables:

table1 (id, minValue)
table2 (table1Id, value)

If I run the query:

UPDATE table1 SET minValue = (SELECT MIN(value) FROM table2 WHERE table1.id == table2.table1Id);

I get the expected values. However if I run this query the minValues remain null:

UPDATE table1 SET minValue = MIN(minValue, (SELECT MIN(value) FROM table2 WHERE table1.id == table2.table1Id));

I will be updating the minValue from multiple tables so I want to make sure it stays smaller than the incoming values but still gets updated when minValue is still null.