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