Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι → ι be given.
Assume H0: ∀ x3 . x3 ∈ x0 ⟶ x1 x3 ⊆ x2 x3.
Let x3 of type ι be given.
Assume H1:
x3 ∈ Pi x0 (λ x4 . x1 x4).
Apply PiE with
x0,
x1,
x3,
x3 ∈ Pi x0 (λ x4 . x2 x4) leaving 2 subgoals.
The subproof is completed by applying H1.
Assume H2:
∀ x4 . x4 ∈ x3 ⟶ and (pair_p x4) (ap x4 0 ∈ x0).
Assume H3:
∀ x4 . x4 ∈ x0 ⟶ ap x3 x4 ∈ x1 x4.
Apply PiI with
x0,
x2,
x3 leaving 2 subgoals.
The subproof is completed by applying H2.
Let x4 of type ι be given.
Assume H4: x4 ∈ x0.
Apply H0 with
x4,
ap x3 x4 leaving 2 subgoals.
The subproof is completed by applying H4.
Apply H3 with
x4.
The subproof is completed by applying H4.