simp [u', Finset.mem_imageOfFinset_iff]
case a
L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝¹ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Finset (Sentence (languageFinset u)) := u.imageOfFinset fun x hσ => toSubLanguageFinsetSelf hσ
τ : Semiformula L Empty 0
⊢ τ ∈ Finset.image (⇑(lMap L.ofSubLanguage)) u' ↔ τ ∈ u
case a
L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝¹ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Finset (Sentence (languageFinset u)) := u.imageOfFinset fun x hσ => toSubLanguageFinsetSelf hσ
τ : Semiformula L Empty 0
⊢ (∃ a, (∃ a_1, ∃ (ha : a_1 ∈ u), toSubLanguageFinsetSelf ha = a) ∧ (lMap L.ofSubLanguage) a = τ) ↔ τ ∈ u