N1
を前提に
をぜんていに
On the premise of
娘さんとは結婚を前提にお付き合いをさせていただいています。
I am dating your daughter on the premise of marriage.
Explanation
「Nを前提に」 means 'on the premise of N'. What follows is built on this condition; if the condition fails, the rest loses its point: 「私たちは、結婚を前提に真剣に交際している」 (We are seriously dating with marriage in mind), 「このシステムは、社内ネットワークでの利用を前提に作られている」 (This system is designed on the premise that it is used on the company network). 「を前提として」 also works. Before a noun use 「〜を前提とした+noun」.
Mixed up with
「〜を前提に」: a required precondition; if it disappears, what follows no longer applies.
「〜を契機に」: a trigger or turning point that starts a change.
「〜をベースに / をもとに」: using some material, model or data as the base for creating or developing.