Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι → ι be given.
Apply set_ext with
{x2 x3|x3 ∈ {x1 x3|x3 ∈ x0}},
{x2 (x1 x3)|x3 ∈ x0} leaving 2 subgoals.
Let x3 of type ι be given.
Assume H0: x3 ∈ {x2 x4|x4 ∈ {x1 x4|x4 ∈ x0}}.
Apply ReplE_impred with
{x1 x4|x4 ∈ x0},
x2,
x3,
x3 ∈ {x2 (x1 x4)|x4 ∈ x0} leaving 2 subgoals.
The subproof is completed by applying H0.
Let x4 of type ι be given.
Assume H1: x4 ∈ {x1 x5|x5 ∈ x0}.
Assume H2: x3 = x2 x4.
Apply ReplE_impred with
x0,
x1,
x4,
x3 ∈ {x2 (x1 x5)|x5 ∈ x0} leaving 2 subgoals.
The subproof is completed by applying H1.
Let x5 of type ι be given.
Assume H3: x5 ∈ x0.
Assume H4: x4 = x1 x5.
Apply H2 with
λ x6 x7 . x7 ∈ {x2 (x1 x8)|x8 ∈ x0}.
Apply H4 with
λ x6 x7 . x2 x7 ∈ {x2 (x1 x8)|x8 ∈ x0}.
Apply ReplI with
x0,
λ x6 . x2 (x1 x6),
x5.
The subproof is completed by applying H3.
Let x3 of type ι be given.
Assume H0: x3 ∈ {x2 (x1 x4)|x4 ∈ x0}.
Apply ReplE_impred with
x0,
λ x4 . x2 (x1 x4),
x3,
x3 ∈ prim5 (prim5 x0 x1) x2 leaving 2 subgoals.
The subproof is completed by applying H0.
Let x4 of type ι be given.
Assume H1: x4 ∈ x0.
Assume H2: x3 = x2 (x1 x4).
Apply H2 with
λ x5 x6 . x6 ∈ {x2 x7|x7 ∈ {x1 x7|x7 ∈ x0}}.
Apply ReplI with
{x1 x5|x5 ∈ x0},
x2,
x1 x4.
Apply ReplI with
x0,
x1,
x4.
The subproof is completed by applying H1.