Let x0 of type ι be given.
Let x1 of type ι → ο be given.
Let x2 of type ι → ι be given.
Let x3 of type ι be given.
Assume H0: x3 ∈ {x2 x4|x4 ∈ {x4 ∈ x0|x1 x4}}.
Apply ReplE with
{x4 ∈ x0|x1 x4},
x2,
x3,
∃ x4 . and (and (x4 ∈ x0) (x1 x4)) (x3 = x2 x4) leaving 2 subgoals.
The subproof is completed by applying H0.
Let x4 of type ι be given.
Assume H1:
and (x4 ∈ {x5 ∈ x0|x1 x5}) (x3 = x2 x4).
Apply H1 with
∃ x5 . and (and (x5 ∈ x0) (x1 x5)) (x3 = x2 x5).
Assume H2: x4 ∈ {x5 ∈ x0|x1 x5}.
Assume H3: x3 = x2 x4.
Apply SepE with
x0,
x1,
x4,
∃ x5 . and (and (x5 ∈ x0) (x1 x5)) (x3 = x2 x5) leaving 2 subgoals.
The subproof is completed by applying H2.
Assume H4: x4 ∈ x0.
Assume H5: x1 x4.
Let x5 of type ο be given.
Assume H6:
∀ x6 . and (and (x6 ∈ x0) (x1 x6)) (x3 = x2 x6) ⟶ x5.
Apply H6 with
x4.
Apply and3I with
x4 ∈ x0,
x1 x4,
x3 = x2 x4 leaving 3 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
The subproof is completed by applying H3.