-- Mathlib/ModelTheory/Fraisse.lean
theorem exists_cg_is_age_of (hn : K.Nonempty)
    (hc : (Quotient.mk' '' K).Countable)
    (fg : ∀ M : Bundled.{w} L.Structure, M ∈ K → Structure.FG L M) (hp : Hereditary K)
    (jep : JointEmbedding K) : ∃ M : Bundled.{w} L.Structure, Structure.CG L M ∧ L.age M = K

Proof state:
case intro
L : Language
K : Set (Bundled L.Structure)
hn : K.Nonempty
hc : (Quotient.mk' '' K).Countable
fg : ∀ M ∈ K, Structure.FG L ↑M
hp : Hereditary K
jep : JointEmbedding K
F : ℕ → Quotient equivSetoid
hF : ∀ (a : Bundled L.Structure), (∃ x ∈ K, (Quotient.mk' x).out ≈ a) ↔ ∃ y, (F y).out ≈ a
⊢ ∃ M, Structure.CG L ↑M ∧ L.age ↑M = K

Tactic:
have hF' : ∀ n : ℕ, (F n).out ∈ K := by
  intro n
  obtain ⟨P, hP1, hP2⟩ := (hF (F n).out).2 ⟨n, Setoid.refl _⟩
  replace hP2 := Setoid.trans (Setoid.symm (Quotient.mk_out P)) hP2
  exact (hp.is_equiv_invariant_of_fg fg _ _ hP2).1 hP1

Result:
case intro
L : Language
K : Set (Bundled L.Structure)
hn : K.Nonempty
hc : (Quotient.mk' '' K).Countable
fg : ∀ M ∈ K, Structure.FG L ↑M
hp : Hereditary K
jep : JointEmbedding K
F : ℕ → Quotient equivSetoid
hF : ∀ (a : Bundled L.Structure), (∃ x ∈ K, (Quotient.mk' x).out ≈ a) ↔ ∃ y, (F y).out ≈ a
hF' : ∀ (n : ℕ), (F n).out ∈ K
⊢ ∃ M, Structure.CG L ↑M ∧ L.age ↑M = K

Proof state:
case intro
L : Language
K : Set (Bundled L.Structure)
hn : K.Nonempty
hc : (Quotient.mk' '' K).Countable
fg : ∀ M ∈ K, Structure.FG L ↑M
hp : Hereditary K
jep : JointEmbedding K
F : ℕ → Quotient equivSetoid
hF : ∀ (a : Bundled L.Structure), (∃ x ∈ K, (Quotient.mk' x).out ≈ a) ↔ ∃ y, (F y).out ≈ a
hF' : ∀ (n : ℕ), (F n).out ∈ K
⊢ ∃ M, Structure.CG L ↑M ∧ L.age ↑M = K

Tactic:
choose P hPK hP hFP using fun (N : K) (n : ℕ) => jep N N.2 (F (n + 1)).out (hF' _)

Result:
case intro
L : Language
K : Set (Bundled L.Structure)
hn : K.Nonempty
hc : (Quotient.mk' '' K).Countable
fg : ∀ M ∈ K, Structure.FG L ↑M
hp : Hereditary K
jep : JointEmbedding K
F : ℕ → Quotient equivSetoid
hF : ∀ (a : Bundled L.Structure), (∃ x ∈ K, (Quotient.mk' x).out ≈ a) ↔ ∃ y, (F y).out ≈ a
hF' : ∀ (n : ℕ), (F n).out ∈ K
P : ↑K → ℕ → Bundled L.Structure
hPK : ∀ (N : ↑K) (n : ℕ), P N n ∈ K
hP : ∀ (N : ↑K) (n : ℕ), (fun M N => Nonempty (↑M ↪[L] ↑N)) (↑N) (P N n)
hFP : ∀ (N : ↑K) (n : ℕ), (fun M N => Nonempty (↑M ↪[L] ↑N)) (F (n + 1)).out (P N n)
⊢ ∃ M, Structure.CG L ↑M ∧ L.age ↑M = K

Proof state:
case intro.refine_2.intro.intro.succ
L : Language
K : Set (Bundled L.Structure)
hn : K.Nonempty
hc : (Quotient.mk' '' K).Countable
fg : ∀ M ∈ K, Structure.FG L ↑M
hp : Hereditary K
jep : JointEmbedding K
F : ℕ → Quotient equivSetoid
hF : ∀ (a : Bundled L.Structure), (∃ x ∈ K, (Quotient.mk' x).out ≈ a) ↔ ∃ y, (F y).out ≈ a
hF' : ∀ (n : ℕ), (F n).out ∈ K
P : ↑K → ℕ → Bundled L.Structure
hPK : ∀ (N : ↑K) (n : ℕ), P N n ∈ K
hP : ∀ (N : ↑K) (n : ℕ), (fun M N => Nonempty (↑M ↪[L] ↑N)) (↑N) (P N n)
hFP : ∀ (N : ↑K) (n : ℕ), (fun M N => Nonempty (↑M ↪[L] ↑N)) (F (n + 1)).out (P N n)
G : ℕ → ↑K := Nat.rec ⟨(F 0).out, ⋯⟩ fun n N => ⟨P N n, ⋯⟩
f : (i j : ℕ) → i ≤ j → ↑↑(G i) ↪[L] ↑↑(G j) := DirectedSystem.natLERec (id fun n => Nonempty.some ⋯)
this✝ : DirectedSystem (fun n => ↑↑(G n)) fun i j h => ⇑(f i j h)
N : Bundled L.Structure
KN : N ∈ K
this : (Quotient.mk' N).out ≈ N
n : ℕ
e : ↑(F (n + 1)).out ≃[L] ↑N
⊢ ↑(F (n + 1)).out ↪[L] ↑↑(G (n + 1))

Tactic:
dsimp [G]

Result:
case intro.refine_2.intro.intro.succ
L : Language
K : Set (Bundled L.Structure)
hn : K.Nonempty
hc : (Quotient.mk' '' K).Countable
fg : ∀ M ∈ K, Structure.FG L ↑M
hp : Hereditary K
jep : JointEmbedding K
F : ℕ → Quotient equivSetoid
hF : ∀ (a : Bundled L.Structure), (∃ x ∈ K, (Quotient.mk' x).out ≈ a) ↔ ∃ y, (F y).out ≈ a
hF' : ∀ (n : ℕ), (F n).out ∈ K
P : ↑K → ℕ → Bundled L.Structure
hPK : ∀ (N : ↑K) (n : ℕ), P N n ∈ K
hP : ∀ (N : ↑K) (n : ℕ), (fun M N => Nonempty (↑M ↪[L] ↑N)) (↑N) (P N n)
hFP : ∀ (N : ↑K) (n : ℕ), (fun M N => Nonempty (↑M ↪[L] ↑N)) (F (n + 1)).out (P N n)
G : ℕ → ↑K := Nat.rec ⟨(F 0).out, ⋯⟩ fun n N => ⟨P N n, ⋯⟩
f : (i j : ℕ) → i ≤ j → ↑↑(G i) ↪[L] ↑↑(G j) := DirectedSystem.natLERec (id fun n => Nonempty.some ⋯)
this✝ : DirectedSystem (fun n => ↑↑(G n)) fun i j h => ⇑(f i j h)
N : Bundled L.Structure
KN : N ∈ K
this : (Quotient.mk' N).out ≈ N
n : ℕ
e : ↑(F (n + 1)).out ≃[L] ↑N
⊢ ↑(F (n + 1)).out ↪[L] ↑(P (Nat.rec ⟨(F 0).out, ⋯⟩ (fun n N => ⟨P N n, ⋯⟩) n) n)

Proof state:
case intro
L : Language
K : Set (Bundled L.Structure)
hn : K.Nonempty
hc : (Quotient.mk' '' K).Countable
fg : ∀ M ∈ K, Structure.FG L ↑M
hp : Hereditary K
jep : JointEmbedding K
F : ℕ → Quotient equivSetoid
hF : Quotient.mk' '' K = range F
⊢ ∃ M, Structure.CG L ↑M ∧ L.age ↑M = K

Tactic:
simp only [Set.ext_iff, Quotient.forall, mem_image, mem_range, Quotient.eq'] at hF

Result:
case intro
L : Language
K : Set (Bundled L.Structure)
hn : K.Nonempty
hc : (Quotient.mk' '' K).Countable
fg : ∀ M ∈ K, Structure.FG L ↑M
hp : Hereditary K
jep : JointEmbedding K
F : ℕ → Quotient equivSetoid
hF : ∀ (a : Bundled L.Structure), (∃ x ∈ K, Quotient.mk' x = ⟦a⟧) ↔ ∃ y, F y = ⟦a⟧
⊢ ∃ M, Structure.CG L ↑M ∧ L.age ↑M = K

Proof state:
case intro.refine_2.intro.intro
L : Language
K : Set (Bundled L.Structure)
hn : K.Nonempty
hc : (Quotient.mk' '' K).Countable
fg : ∀ M ∈ K, Structure.FG L ↑M
hp : Hereditary K
jep : JointEmbedding K
F : ℕ → Quotient equivSetoid
hF : ∀ (a : Bundled L.Structure), (∃ x ∈ K, (Quotient.mk' x).out ≈ a) ↔ ∃ y, (F y).out ≈ a
hF' : ∀ (n : ℕ), (F n).out ∈ K
P : ↑K → ℕ → Bundled L.Structure
hPK : ∀ (N : ↑K) (n : ℕ), P N n ∈ K
hP : ∀ (N : ↑K) (n : ℕ), (fun M N => Nonempty (↑M ↪[L] ↑N)) (↑N) (P N n)
hFP : ∀ (N : ↑K) (n : ℕ), (fun M N => Nonempty (↑M ↪[L] ↑N)) (F (n + 1)).out (P N n)
G : ℕ → ↑K := Nat.rec ⟨(F 0).out, ⋯⟩ fun n N => ⟨P N n, ⋯⟩
f : (i j : ℕ) → i ≤ j → ↑↑(G i) ↪[L] ↑↑(G j) := DirectedSystem.natLERec (id fun n => Nonempty.some ⋯)
this✝ : DirectedSystem (fun n => ↑↑(G n)) fun i j h => ⇑(f i j h)
N : Bundled L.Structure
KN : N ∈ K
this : (Quotient.mk' N).out ≈ N
n : ℕ
e : ↑(F n).out ≃[L] ↑N
⊢ ↑(F n).out ↪[L] ↑↑(G n)

Tactic:
rcases n with - | n

Result:
case intro.refine_2.intro.intro.zero
L : Language
K : Set (Bundled L.Structure)
hn : K.Nonempty
hc : (Quotient.mk' '' K).Countable
fg : ∀ M ∈ K, Structure.FG L ↑M
hp : Hereditary K
jep : JointEmbedding K
F : ℕ → Quotient equivSetoid
hF : ∀ (a : Bundled L.Structure), (∃ x ∈ K, (Quotient.mk' x).out ≈ a) ↔ ∃ y, (F y).out ≈ a
hF' : ∀ (n : ℕ), (F n).out ∈ K
P : ↑K → ℕ → Bundled L.Structure
hPK : ∀ (N : ↑K) (n : ℕ), P N n ∈ K
hP : ∀ (N : ↑K) (n : ℕ), (fun M N => Nonempty (↑M ↪[L] ↑N)) (↑N) (P N n)
hFP : ∀ (N : ↑K) (n : ℕ), (fun M N => Nonempty (↑M ↪[L] ↑N)) (F (n + 1)).out (P N n)
G : ℕ → ↑K := Nat.rec ⟨(F 0).out, ⋯⟩ fun n N => ⟨P N n, ⋯⟩
f : (i j : ℕ) → i ≤ j → ↑↑(G i) ↪[L] ↑↑(G j) := DirectedSystem.natLERec (id fun n => Nonempty.some ⋯)
this✝ : DirectedSystem (fun n => ↑↑(G n)) fun i j h => ⇑(f i j h)
N : Bundled L.Structure
KN : N ∈ K
this : (Quotient.mk' N).out ≈ N
e : ↑(F 0).out ≃[L] ↑N
⊢ ↑(F 0).out ↪[L] ↑↑(G 0)

case intro.refine_2.intro.intro.succ
L : Language
K : Set (Bundled L.Structure)
hn : K.Nonempty
hc : (Quotient.mk' '' K).Countable
fg : ∀ M ∈ K, Structure.FG L ↑M
hp : Hereditary K
jep : JointEmbedding K
F : ℕ → Quotient equivSetoid
hF : ∀ (a : Bundled L.Structure), (∃ x ∈ K, (Quotient.mk' x).out ≈ a) ↔ ∃ y, (F y).out ≈ a
hF' : ∀ (n : ℕ), (F n).out ∈ K
P : ↑K → ℕ → Bundled L.Structure
hPK : ∀ (N : ↑K) (n : ℕ), P N n ∈ K
hP : ∀ (N : ↑K) (n : ℕ), (fun M N => Nonempty (↑M ↪[L] ↑N)) (↑N) (P N n)
hFP : ∀ (N : ↑K) (n : ℕ), (fun M N => Nonempty (↑M ↪[L] ↑N)) (F (n + 1)).out (P N n)
G : ℕ → ↑K := Nat.rec ⟨(F 0).out, ⋯⟩ fun n N => ⟨P N n, ⋯⟩
f : (i j : ℕ) → i ≤ j → ↑↑(G i) ↪[L] ↑↑(G j) := DirectedSystem.natLERec (id fun n => Nonempty.some ⋯)
this✝ : DirectedSystem (fun n => ↑↑(G n)) fun i j h => ⇑(f i j h)
N : Bundled L.Structure
KN : N ∈ K
this : (Quotient.mk' N).out ≈ N
n : ℕ
e : ↑(F (n + 1)).out ≃[L] ↑N
⊢ ↑(F (n + 1)).out ↪[L] ↑↑(G (n + 1))