John Calvin, in his commentary on John, shows how Total Depravity and Irresistible Grace logically (and obviously theologically) are inseparable. This is what Calvin had to say on the drawing that the Father does in John 6:
“It is not violent, so as to compel men by external force; but still it is a powerful impulse of the Holy Spirit, which makes men willing who formerly were unwilling and reluctant. It is a false and profane assertion, therefore that none are drawn but those who are willing to be drawn, as if man made himself obedient to God by His own efforts; for the willingness with which men follow God is what they have already have from Himself, who has formed their hearts to obey Him.” ~John Calvin (Calvin’s Commentaries Vol. XVII)
There is nothing inherent in our hearts that enables us to be drawn to the Father. It is completely His work; this work is a beautiful, glorious, and powerful act of grace. He is not dragging us, kicking and screaming into heaven; rather, He calls us forth as Lazarus from the tomb, for when He says “Come forth!”, our heart of stone shatters into the dust of the fallen Adam, and a new man is born with a heart of flesh into the New Adam.