Let x0 of type ι be given.
Let x1 of type ι → ι → ι be given.
Assume H0: ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ∈ x0.
Assume H1:
∃ x2 . and (x2 ∈ x0) (∀ x3 . x3 ∈ x0 ⟶ and (x1 x3 x2 = x3) (x1 x2 x3 = x3)).
Assume H2:
∀ x2 . x2 ∈ x0 ⟶ bij x0 x0 (λ x3 . x1 x2 x3).
Assume H3:
∀ x2 . x2 ∈ x0 ⟶ bij x0 x0 (λ x3 . x1 x3 x2).
Apply andI with
struct_b (pack_b x0 x1),
unpack_b_o (pack_b x0 x1) (λ x2 . λ x3 : ι → ι → ι . and (and (∃ x4 . and (x4 ∈ x2) (∀ x5 . x5 ∈ x2 ⟶ and (x3 x5 x4 = x5) (x3 x4 x5 = x5))) (∀ x4 . x4 ∈ x2 ⟶ bij x2 x2 (λ x5 . x3 x4 x5))) (∀ x4 . x4 ∈ x2 ⟶ bij x2 x2 (λ x5 . x3 x5 x4))) leaving 2 subgoals.
Apply pack_struct_b_I with
x0,
x1.
The subproof is completed by applying H0.
Apply unknownprop_79c8405166f8f53d313e9b10a06ea6c3dfc3e70be1e0a2be4c457f6dd42e2d2d with
x0,
x1,
λ x2 x3 : ο . x3.
Apply and3I with
∃ x2 . and (x2 ∈ x0) (∀ x3 . x3 ∈ x0 ⟶ and (x1 x3 x2 = x3) (x1 x2 x3 = x3)),
∀ x2 . x2 ∈ x0 ⟶ bij x0 x0 (λ x3 . x1 x2 x3),
∀ x2 . x2 ∈ x0 ⟶ bij x0 x0 (λ x3 . x1 x3 x2) leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying H3.