Many things that Jesus spoke and did were in fulfillment of the law of Moses, and in the prophets, and in the psalms concerning Him.

"And he said unto them, These are the words which I spake unto you, while I was yet with you, that all things must be fulfilled, which were written in the law of Moses, and in the prophets, and in the psalms, concerning me. Then opened he their understanding, that they might understand the scriptures," - Luke 24:44-45 KJV

"Philip findeth Nathanael, and saith unto him, We have found him, of whom Moses in the law, and the prophets, did write, Jesus of Nazareth, the son of Joseph." - John 1:45 KJV

Moses foretold that God would raise up a prophet like himself.

"For Moses truly said unto the fathers, A prophet shall the Lord your God raise up unto you of your brethren, like unto me; him shall ye hear in all things whatsoever he shall say unto you." - Acts 3:22 KJV

"Then those men, when they had seen the miracle that Jesus did, said, This is of a truth that prophet that should come into the world." - John 6:14 KJV

Moses inaugurated Aaron and his sons into the priesthood first by washing them. The law describes the washing of the priests at the golden laver.

"When they go into the tabernacle of the congregation, they shall wash with water, that they die not; or when they come near to the altar to minister, to burn offering made by fire unto the LORD: so they shall wash their hands and their feet, that they die not: and it shall be a statute for ever to them, even to him and to his seed throughout their generations." - Exodus 30:20-21 KJV

When Moses inaugurated Aaron and his sons into the priesthood, God commanded him to wash them.

"And Moses did as the LORD commanded him; and the assembly was gathered together unto the door of the tabernacle of the congregation. And Moses said unto the congregation, This is the thing which the LORD commanded to be done. And Moses brought Aaron and his sons, and washed them with water." - Leviticus 8:4-6 KJV

He then clothed them with their priest’s clothing, anointed them, and then made offerings for them. Finally, Moses applied blood from the altar on Aaron and his sons and on their garments, then instructed them boil the remaining meat offerings at the Tabernacle entrance, and eat it there, along with the bread that is in the basket of offerings for their consecration.

Just as Moses inaugurated Aaron and his sons into the priesthood of the Old Covenant of the Law, Jesus inaugurated His disciples into His new priesthood of the New Covenant of mercy, grace, and servanthood.

"To whom coming, as unto a living stone, disallowed indeed of men, but chosen of God, and precious, ye also, as lively stones, are built up a spiritual house, an holy priesthood, to offer up spiritual sacrifices, acceptable to God by Jesus Christ. ... But ye are a chosen generation, a royal priesthood, an holy nation, a peculiar people; that ye should shew forth the praises of him who hath called you out of darkness into his marvellous light: which in time past were not a people, but are now the people of God: which had not obtained mercy, but now have obtained mercy." - 1 Peter 2:4-5, 9-10 KJV

It appears Jesus inaugurated this new priesthood the night of the last supper.

"He riseth from supper, and laid aside his garments; and took a towel, and girded himself. After that he poureth water into a bason, and began to wash the disciples' feet, and to wipe them with the towel wherewith he was girded. Then cometh he to Simon Peter: and Peter saith unto him, Lord, dost thou wash my feet? Jesus answered and said unto him, What I do thou knowest not now; but thou shalt know hereafter. Peter saith unto him, Thou shalt never wash my feet. Jesus answered him, If I wash thee not, thou hast no part with me." - John 13:4-8 KJV

"And as they did eat, Jesus took bread, and blessed, and brake it, and gave to them, and said, Take, eat: this is my body. And he took the cup, and when he had given thanks, he gave it to them: and they all drank of it. And he said unto them, This is my blood of the new testament [new covenant], which is shed for many." - Mark 14:22-24 KJV