The following will be shown: Let I be a σ-ideal on a Polish space X so that the associated forcing of I+${\bf{\Delta }}_1^1$ sets ordered by ⊆ is a proper forcing. Let E be a ${\bf{\Sigma }}_1^1$ or a ${\bf{\Pi }}_1^1$ equivalence relation on X with all equivalence classes ${\bf{\Delta }}_1^1$. If for all $z \in {H_{{{\left}^ + }}}$, z♯ exists, then there exists an I+${\bf{\Delta }}_1^1$ set C ⊆ X such that E ↾ C is a ${\bf{\Delta }}_1^1$ equivalence (...) relation. (shrink)

If the bounded proper forcing axiom BPFA holds and ω 1 = ${\mathrm{\omega }}_{1}^{\mathrm{L}}$ , then there is a lightface ${\mathrm{\Sigma }}_{3}^{1}$ well-ordering of the reals. The argument combines a well-ordering due to Caicedo-Veličković with an absoluteness result for models of MA in the spirit of "David's trick." We also present a general coding scheme that allows us to show that BPFA is equiconsistent with R being lightface ${\mathrm{\Sigma }}_{4}^{1}$ , for many "consistently locally certified" relations R on $\mathrm{\mathbb{R}}$ . (...) This is accomplished through a use of David's trick and a coding through the ∑ 2 stable ordinals of L. (shrink)