At John 13:1-30 we see the narrative of the last Passover meal that Jesus partook with his disciples, in which Judas is identified as the one who would betray him :
.... After saying this Jesus was troubled in spirit, and declared, “Very truly, I tell you, one of you will betray me.” .... So while reclining next to Jesus, he (John ) asked him, “Lord, who is it?” . Jesus answered, “It is the one to whom I give this piece of bread when I have dipped it in the dish.” So when he had dipped the piece of bread, he gave it to Judas son of Simon Iscariot. After he received the piece of bread, Satan entered into him. Jesus said to him, “Do quickly what you are going to do.” ... So, after receiving the piece of bread, he immediately went out. And it was night
Now, the Gospel according John does not contain an explcit narration of the Institution of Holy Eucharist . Reading that with I Corinthians 11:25 which says that the supper had been ended as Jesus took the cup calling it the new covenant of His Blood, one is inclined to believe that Judas had left the venue even before the Eucharist was constituted.
My question therefore, is: Does the Catholic Church categorically teach that Judas Iscariot participated in the First Eucharist instituted by Jesus at the Last Supper? If it does not, why do the images of Last Supper that we have, contain the picture of twelve Apostles?