Let x0 of type ι → ο be given.
Let x1 of type ι → ι be given.
Let x2 of type ι → ι be given.
Assume H0: ∀ x3 . x0 x3 ⟶ x2 (x1 x3) = x3.
Let x3 of type ι be given.
Assume H1: ∀ x4 . x4 ∈ x3 ⟶ x0 x4.
Apply set_ext with
{x2 x4|x4 ∈ {x1 x4|x4 ∈ x3}},
x3 leaving 2 subgoals.
Let x4 of type ι be given.
Assume H2: x4 ∈ {x2 x5|x5 ∈ {x1 x5|x5 ∈ x3}}.
Apply ReplE_impred with
{x1 x5|x5 ∈ x3},
x2,
x4,
x4 ∈ x3 leaving 2 subgoals.
The subproof is completed by applying H2.
Let x5 of type ι be given.
Assume H3: x5 ∈ {x1 x6|x6 ∈ x3}.
Assume H4: x4 = x2 x5.
Apply ReplE_impred with
x3,
x1,
x5,
x4 ∈ x3 leaving 2 subgoals.
The subproof is completed by applying H3.
Let x6 of type ι be given.
Assume H5: x6 ∈ x3.
Assume H6: x5 = x1 x6.
Apply H4 with
λ x7 x8 . x8 ∈ x3.
Apply H6 with
λ x7 x8 . x2 x8 ∈ x3.
Apply H0 with
x6,
λ x7 x8 . x8 ∈ x3 leaving 2 subgoals.
Apply H1 with
x6.
The subproof is completed by applying H5.
The subproof is completed by applying H5.
Let x4 of type ι be given.
Assume H2: x4 ∈ x3.
Apply H0 with
x4,
λ x5 x6 . x5 ∈ prim5 (prim5 x3 x1) x2 leaving 2 subgoals.
Apply H1 with
x4.
The subproof is completed by applying H2.
Apply ReplI with
{x1 x5|x5 ∈ x3},
x2,
x1 x4.
Apply ReplI with
x3,
x1,
x4.
The subproof is completed by applying H2.