Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι → ι be given.
Assume H0: ∀ x3 . x3 ∈ x0 ⟶ x1 (x2 x3) = x3.
Apply set_ext with
{x1 x3|x3 ∈ {x2 x3|x3 ∈ x0}},
x0 leaving 2 subgoals.
Let x3 of type ι be given.
Assume H1: x3 ∈ {x1 x4|x4 ∈ {x2 x4|x4 ∈ x0}}.
Apply ReplE_impred with
{x2 x4|x4 ∈ x0},
x1,
x3,
x3 ∈ x0 leaving 2 subgoals.
The subproof is completed by applying H1.
Let x4 of type ι be given.
Assume H2: x4 ∈ {x2 x5|x5 ∈ x0}.
Assume H3: x3 = x1 x4.
Apply ReplE_impred with
x0,
x2,
x4,
x3 ∈ x0 leaving 2 subgoals.
The subproof is completed by applying H2.
Let x5 of type ι be given.
Assume H4: x5 ∈ x0.
Assume H5: x4 = x2 x5.
Apply H3 with
λ x6 x7 . x7 ∈ x0.
Apply H5 with
λ x6 x7 . x1 x7 ∈ x0.
Apply H0 with
x5,
λ x6 x7 . x7 ∈ x0 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H4.
Let x3 of type ι be given.
Assume H1: x3 ∈ x0.
Apply H0 with
x3,
λ x4 x5 . x4 ∈ prim5 (prim5 x0 x2) x1 leaving 2 subgoals.
The subproof is completed by applying H1.
Apply ReplI with
{x2 x4|x4 ∈ x0},
x1,
x2 x3.
Apply ReplI with
x0,
x2,
x3.
The subproof is completed by applying H1.