John 15:1-6 (NKJV)
1 "I am the true vine, and My Father is the vinedresser.
2 Every branch in Me that does not bear fruit He takes away; and every branch that bears fruit He prunes, that it may bear more fruit.
3 You are already clean because of the word which I have spoken to you.
4 Abide in Me, and I in you. As the branch cannot bear fruit of itself, unless it abides in the vine, neither can you, unless you abide in Me.
5 I am the vine, you are the branches. He who abides in Me, and I in him, bears much fruit; for without Me you can do nothing.
6 If anyone does not abide in Me, he is cast out as a branch and is withered; and they gather them and throw them into the fire, and they are burned.
A grapevine branch can survive and produce foliage for a while after it has been severed, but it cannot produce fruit unless it is connected to a root stock. As Jesus had a living dependence on the Father, [see John 6:47] so believers in Jesus need to have a living dependence on him.
John 6:47 (NKJV)
47 Most assuredly, I say to you, he who believes in Me has everlasting life.
(He who believes, not he who has fallen away, he who no longer believes, 'Most Assuredly' Jesus says)
They cannot bear fruit without being connected to the vine.
We cannot bear fruit, or remain in the vine unless Jesus is in us, unless we abide in Him and He in us.
Therefore, it is essential that they completely trust Jesus, or they will not be connected to the source of life. [John 15:6]
John 6:53-58 (NKJV)
53 Then Jesus said to them, "Most assuredly, I say to you, unless you eat the flesh of the Son of Man and drink His blood, you have no life in you.
54 Whoever eats My flesh and drinks My blood has eternal life, and I will raise him up at the last day.
55 For My flesh is food indeed, and My blood is drink indeed.
56 He who eats My flesh and drinks My blood abides in Me, and I in him.
57 As the living Father sent Me, and I live because of the Father, so he who feeds on Me will live because of Me.
58 This is the bread which came down from heaven--not as your fathers ate the manna, and are dead. He who eats this bread will live forever."
For a person who has fallen away from our Lord, he no longer abides in Him so Jesus cannot abide in the person.
The person who was believed but now does not no longer abides in Christ, he/she 'no longer eats My flesh and drinks My blood', says Jesus, he/she is no longer fruitful and is pruned from the vine, 'cut off' from the life giving source!
John 15:1-2 (NKJV)
1 "I am the true vine, and My Father is the vinedresser.
2 Every branch in Me that does not bear fruit He takes away; and every branch that bears fruit He prunes, that it may bear more fruit.
John 15:6 (NKJV)
6 If anyone does not abide in Me, he is cast out as a branch and is withered; and they gather them and throw them into the fire, and they are burned.