From a commentary on the gospel of John by Saint Cyril of Alexandria, bishop
I am the vine, you are the branches
The Lord calls himself the vine and those united to him branches in order to teach us how much we shall benefit from our union with him, and how important it is for us to remain in his love. By receiving the Holy Spirit, who is the bond of union between us and Christ our Saviour, those who are joined to him, as branches are to a vine, share in his own nature.
On the part of those who come to the vine, their union with him depends upon a deliberate act of the will; on his part, the union is effected by grace. Because we had good will, we made the act of faith that brought us to Christ, and received from him the dignity of adoptive sonship that made us his own kinsmen, according to the words of Saint Paul: He who is joined to the Lord is one spirit with him.
The prophet Isaiah calls Christ the foundation, because it is upon him that we as living and spiritual stones are built into a holy priesthood to be a dwelling place for God in the Spirit. Upon no other foundation than Christ can this temple be built. Here Christ is teaching the same truth by calling himself the vine, since the vine is the parent of its branches, and provides their nourishment.