Los schema for the universe ultrapower

Fix the formula externally. Atomic equality and membership are the coordinate clauses of F1. Negation complements the truth set, and conjunction intersects two truth sets. A proper ultrafilter contains exactly one of a set and its complement, and contains an intersection exactly when it contains both factors. These prove the Boolean induction steps, as in F2, without using satisfaction for a proper-class structure.

Suppose the coordinate truth set $A=\{i:\mathrm{\exists }x\psi (x,f_1(i),\dots ,f_n(i))\}$ lies in U. For each i in A there is a least ordinal ${\rho }_i$ which is the rank of a witness: first bound the search by the rank of any one witness, then minimize ordinals. This defines rho_i uniquely, so Replacement collects these ordinals. For each i in A, Separation in $V_{{\rho }_i+1}$ gives the nonempty set $W_i$ of witnesses of rank rho_i. Replacement collects the family of W_i, and F3 supplies choices g(i) in W_i. Set g(i) to empty outside A. This is a set function on I. Its matrix truth set contains A, so the induction hypothesis gives a Scott-domain witness [g]. The choices were from sets, not proper classes.

Conversely a Scott-domain existential witness has the form [g] for a set function g. The matrix induction hypothesis says its coordinate matrix truth set belongs to U. This set is contained in the existential truth set, which therefore belongs to U by upward closure. Together with step 2.1 this completes the formula induction. For constant parameters the coordinate truth set is I or empty according to the ambient formula's truth; properness gives preservation and reflection by the constant map. Apply the proved schema to the single axiom of Extensionality, true in V: its coordinate truth set is I, so the Scott structure is extensional. No simultaneous truth definition over all formulas was used.