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