I am wondering how to assess the following modulation from C major to Bb major according to rules of traditional harmony:
I threw the modulation together without thinking about it much, and thought "that'll do", but I want to dig deeper. For context, I'm doing a pop song on piano sort of composition. The song was feeling repetitive, so I wanted to add a bridge that could provide a bit of a break from the otherwise repeating chord progression: I-vi-I-V
I want the transition into the new key to be smooth, so I understand the use of a pivot chord--a chord shared by both keys--would be advisable. D minor and F major are shared by both keys, and, as you can see, I employ the F major chord before the B flat (tonic) arrives. But I know that among the most effective pivot chords are those that would normally lead into the V chord of the new key, such as II and V, but F major is the V chord in Bb major.
Also, I don't know why I chose the D7 chord to precede the F major, because it has a non-diatonic note (the F#).
Basically, I'm lost. I would like to continue working on this modulation, if it makes sense. But I am also happy to totally scrap it if there is a better way to go.