It is immediate that J0 = imageM. If δ > 0, then π−1(δ)< β. Then 〈Jβ, ∈〉 ⊨ (∃x) (∃s)φ(x, π−1(δ), s) and so 〈M, ∈〉 ⊨ (∃x)(∃s)φ(x, δ, s). Let xM and let sM such that 〈M, ∈〉 ⊨ φ(x, δ, s). By the absoluteness of φ(x, δ, s), any admissible structure image containing x, s and δ satisfies φ(x, δ, s). By Proposition 3.3.3, Jδ = xM.

Furthermore, for any xM, there exists a δ < β such that π−1(x)∈ Jδ. Hence 〈Jβ, ∈〉 ⊨ (∃δ)(∃y)(∃s)(π−1(x)∈ yφ(y, δ, s)). Then 〈M, ∈〉 ⊨ (∃δ)(∃y)(∃s)(xyφ(y, δ, s)). By Proposition 3.3.3 again, xJδʹ for some δʹ < ...

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