We present integral field spectroscopic observations of the nearby (z similar to 0.03) dual active galactic nuclei (AGNs) Mrk 739, whose projected nuclear separation is similar to 3.4 kpc, obtained with the Multi Unit Spectroscopic Explorer at the Very Large Telescope. We find that the galaxy has an extended AGN-ionized emission-line region extending up to similar to 20 kpc away from the nuclei, while star-forming regions are more centrally concentrated within 2-3 kpc. We model the kinematics of the ionized gas surrounding the eastern nucleus using a circular disk profile, resulting in a peak velocity of 237(28)(+26) km s(-1) at a distance of similar to 1.2 kpc. The enclosed dynamical mass within 1.2 kpc is logM(M-circle dot) = 10.20 +/- 0.06, similar to 1000 times larger than the estimated supermassive black hole (SMBH) mass of Mrk 739E. The morphology and dynamics of the system are consistent with an early stage of the collision, where the foreground galaxy (Mrk 739W) is a young star-forming galaxy in an ongoing first passage with its background companion (Mrk 739E). Since the SMBH in Mrk 739W does not show evidence of being rapidly accreting, we claim that the northern spiral arms of Mrk 739W are ionized by the nuclear activity of Mrk 739E.