Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Assume H0: ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 = x1 x3 ⟶ x2 = x3.
Apply bijI with
x0,
{x1 x2|x2 ∈ x0},
x1 leaving 3 subgoals.
Let x2 of type ι be given.
Assume H1: x2 ∈ x0.
Apply ReplI with
x0,
x1,
x2.
The subproof is completed by applying H1.
The subproof is completed by applying H0.
Let x2 of type ι be given.
Assume H1: x2 ∈ {x1 x3|x3 ∈ x0}.
Apply ReplE_impred with
x0,
x1,
x2,
∃ x3 . and (x3 ∈ x0) (x1 x3 = x2) leaving 2 subgoals.
The subproof is completed by applying H1.
Let x3 of type ι be given.
Assume H2: x3 ∈ x0.
Assume H3: x2 = x1 x3.
Let x4 of type ο be given.
Assume H4:
∀ x5 . and (x5 ∈ x0) (x1 x5 = x2) ⟶ x4.
Apply H4 with
x3.
Apply andI with
x3 ∈ x0,
x1 x3 = x2 leaving 2 subgoals.
The subproof is completed by applying H2.
Let x5 of type ι → ι → ο be given.
The subproof is completed by applying H3 with λ x6 x7 . x5 x7 x6.