Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι → ι → ο be given.
Let x3 of type ι → ι → ι be given.
Let x4 of type ι be given.
Let x5 of type ο be given.
Assume H1: ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x1 x6 ⟶ x2 x6 x7 ⟶ x4 = x3 x6 x7 ⟶ x5.
Apply UnionE_impred with
{{x3 x6 x7|x7 ∈ x1 x6,x2 x6 x7}|x6 ∈ x0},
x4,
x5 leaving 2 subgoals.
The subproof is completed by applying H0.
Let x6 of type ι be given.
Assume H2: x4 ∈ x6.
Assume H3: x6 ∈ {{x3 x7 x8|x8 ∈ x1 x7,x2 x7 x8}|x7 ∈ x0}.
Apply ReplE_impred with
x0,
λ x7 . {x3 x7 x8|x8 ∈ x1 x7,x2 x7 x8},
x6,
x5 leaving 2 subgoals.
The subproof is completed by applying H3.
Let x7 of type ι be given.
Assume H4: x7 ∈ x0.
Assume H5: x6 = {x3 x7 x8|x8 ∈ x1 x7,x2 x7 x8}.
Claim L6: x4 ∈ {x3 x7 x8|x8 ∈ x1 x7,x2 x7 x8}
Apply H5 with
λ x8 x9 . x4 ∈ x8.
The subproof is completed by applying H2.
Apply ReplSepE_impred with
x1 x7,
x2 x7,
x3 x7,
x4,
x5 leaving 2 subgoals.
The subproof is completed by applying L6.
Let x8 of type ι be given.
Assume H7: x8 ∈ x1 x7.
Assume H8: x2 x7 x8.
Assume H9: x4 = x3 x7 x8.
Apply H1 with
x7,
x8 leaving 4 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
The subproof is completed by applying H9.