The processional and recessional commonly played at weddings are actually by two completely different composers. The anthem 'Treulich geführt' (now known as 'Here Comes the Bride' or the Bridal Chorus) is from Wagner's opera 'Lohengrin'. The original draft is preserved at the Karpeles Manuscript Library Museum.
The recessional, however, is taken from the entr'acte of Mendelssohn's incidental music to A Midsummer Night's Dream. The fashion for playing this as a recessional at weddings originates from a performance of this piece at the wedding of the English Princess Royal after Mendelssohn's death in 1858.