Merging results present in all sub queries - sql

Question

I have the a postgre table with a similar representation to the data below.
Caters Table:
-----------------------
| Name | Option |
-----------------------
| jane | social |
| jane | vegan |
| jane | gmo-free |
| jane | italian |
| jack | social |
| jack | corporate |
| jack | gmo-free |
| jack | greek |
| rodz | social |
| rodz | wedding |
| rodz | gmo-free |
| rodz | vegan |
| rodz | french |
This is the "pseudo" query I'm trying to run
SELECT * FROM caters
WHERE option is either ['italian', 'french']
AND WHERE option is both ['wedding', 'social']
This pseudo query should return rodz. Because it either has italian or french and it has both wedding and social.
This is the query I tried to write to accomplish my sudo query
SELECT c.name FROM caters c
WHERE c.option in ('italian', 'french')
GROUP BY c.name
HAVING array_agg(c.option) #> array['wedding', 'social']
How ever this returns no results. Running the query individually
SELECT c.name FROM caters c
WHERE c.option in ('italian', 'french')
GROUP BY c.name
Result:
-----------
| Name |
-----------
| jane | // has italian
| rodz | // has french
The other query
SELECT c.name FROM caters c
GROUP BY c.name
HAVING array_agg(c.option) #> array['wedding', 'social']
Result:
-----------
| Name |
-----------
| rodz | // has wedding and social
So I can see individually the queries are correct. This made me think well if I have 2 queries giving me the correct results just need to filter out results that are in both queries why don't I JOIN them.
So I tried
SELECT c.name FROM caters c
JOIN caters c1
ON c1.name = c.name and c1.option = c.option
WHERE c1.option in ('italian', 'french')
GROUP BY c.name
HAVING array_agg(c.option) #> array['wedding', 'social']
But this also yielding no results. Any idea how I can go about this?
NOTE: The query is dynamic each time its ran the values being used could be different sometimes maybes it 5 languages sometimes its 2 languages like in this example ('italian', 'french'). To give an example what I mean by dynamic query another query could be
SELECT * FROM caters
WHERE option is either ['italian']
AND WHERE option is both ['corporate', 'social']
// returns none
----------------------------------------------------------
SELECT * FROM caters
WHERE option is either ['french', 'greek']
AND WHERE option is either ['gmo-free', 'vegan']
AND WHERE option is both ['corporate', 'social']
// returns jack
----------------------------------------------------------
SELECT * FROM caters WHERE option is ['social']
// returns jack, and rodz

Answer 1

You can try using a correlated subquery
DEMO
select distinct name from tablename a
where option in ('italian', 'french') and exists
(
select 1 from tablename b where a.name=b.name and option in ('wedding', 'social')
group by b.name having count(distinct option)=2
)
OUTPUT:
name
rodz

Answer 2

Here is one method:
SELECT c.name
FROM caters c
WHERE c.option in ('italian', 'french', 'wedding', 'social')
GROUP BY c.name
HAVING COUNT(*) FILTER (WHERE c.option IN ('italian', 'french')) >= 1 AND
COUNT(*) FILTER (WHERE c.option IN ('wedding', 'social')) = 2;