Many of us--as modern readers rendering an understanding of the ancient agricultural world--will miss a deep revelation in the above passage.
When we think of branches, we think of separate sticks, parts, portions which can exist, or at least fall away, from a tree.
Yet with vines, there is no clear or abrupt distinction between the branches and the vine which holds those branches.
In effect, the branches cannot help but identify completely with the vine, for the branches would not exist.
This issue of identity is crucial to the Christian walk
We are not called to imitate Christ through efforts and physical miming.
We are called to identify with Christ, because we have become partakers of His nature:
"Whereby are given unto us exceeding great and precious promises: that by these ye might be partakers of the divine nature, having escaped the corruption that is in the world through lust." (2 Peter 1:4)
"For ye have not received the spirit of bondage again to fear; but ye have received the Spirit of adoption, whereby we cry, Abba, Father." (Romans 8:15)
"And it shall come to pass, that in the place where it was said unto them, You are not my people; there shall they be called the children of the living God." (Romans 9:26)
Let us not forget that Jesus during His earthly ministry served His Father because He allowed His Father to work through Him:
"17But Jesus answered them, My Father worketh hitherto, and I work.
18Therefore the Jews sought the more to kill him, because he not only had broken the Sabbath, but said also that God was his Father, making himself equal with God.
19Then answered Jesus and said unto them, Verily, verily, I say unto you, The Son can do nothing of himself, but what he seeth the Father do: for what things soever he doeth, these also doeth the Son likewise" (John 5: 17-19)
Notice that Jesus declares: "The Father works"
He did not see Himself operating in His own effort, but fully identifying with His Father, and His Father working in Him!
There are more statements which affirm this blessed dependence demonstrated by Jesus as the Perfect Man walking in this imperfect earth:
"I and my Father are one." (John 10:30)
"Believest thou not that I am in the Father, and the Father in me? the words that I speak unto you I speak not of myself: but the Father that dwelleth in me, he doeth the works." (John 14:10)
Just as Jesus was one with His Father, so today we are one with Jesus!
He is the Vine, and we are completely identified with Him, depending on Him for everything!