Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ο be given.
Assume H1:
∀ x3 . x3 ⊆ x1 ⟶ equip x0 x3 ⟶ x2.
Apply H0 with
x2.
Let x3 of type ι → ι be given.
Apply H2 with
x2.
Assume H3: ∀ x4 . x4 ∈ x0 ⟶ x3 x4 ∈ x1.
Assume H4: ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x3 x4 = x3 x5 ⟶ x4 = x5.
Apply H1 with
{x3 x4|x4 ∈ x0} leaving 2 subgoals.
Let x4 of type ι be given.
Assume H5:
x4 ∈ prim5 x0 x3.
Apply ReplE_impred with
x0,
x3,
x4,
x4 ∈ x1 leaving 2 subgoals.
The subproof is completed by applying H5.
Let x5 of type ι be given.
Assume H6: x5 ∈ x0.
Assume H7: x4 = x3 x5.
Apply H7 with
λ x6 x7 . x7 ∈ x1.
Apply H3 with
x5.
The subproof is completed by applying H6.
Let x4 of type ο be given.
Assume H5:
∀ x5 : ι → ι . bij x0 {x3 x6|x6 ∈ x0} x5 ⟶ x4.
Apply H5 with
x3.
Apply bijI with
x0,
{x3 x5|x5 ∈ x0},
x3 leaving 3 subgoals.
Let x5 of type ι be given.
Assume H6: x5 ∈ x0.
Apply ReplI with
x0,
x3,
x5.
The subproof is completed by applying H6.
The subproof is completed by applying H4.
Let x5 of type ι be given.
Assume H6: x5 ∈ {x3 x6|x6 ∈ x0}.
Apply ReplE_impred with
x0,
x3,
x5,
∃ x6 . and (x6 ∈ x0) (x3 x6 = x5) leaving 2 subgoals.
The subproof is completed by applying H6.
Let x6 of type ι be given.
Assume H7: x6 ∈ x0.
Assume H8: x5 = x3 x6.
Let x7 of type ο be given.
Assume H9:
∀ x8 . and (x8 ∈ x0) (x3 x8 = x5) ⟶ x7.
Apply H9 with
x6.
Apply andI with
x6 ∈ x0,
x3 x6 = x5 leaving 2 subgoals.
The subproof is completed by applying H7.
Let x8 of type ι → ι → ο be given.
The subproof is completed by applying H8 with λ x9 x10 . x8 x10 x9.