Let x0 of type ι → ι → ο be given.
Let x1 of type ι → ι be given.
Assume H0:
∀ x2 . x2 ∈ u12 ⟶ ∀ x3 . x3 ∈ u12 ⟶ x1 x2 = x1 x3 ⟶ x2 = x3.
Let x2 of type ι be given.
Let x3 of type ο be given.
Assume H2:
∀ x4 : ι → ι . bij x2 {x1 x5|x5 ∈ x2} x4 ⟶ x3.
Apply H2 with
x1.
Apply bijI with
x2,
{x1 x4|x4 ∈ x2},
x1 leaving 3 subgoals.
Let x4 of type ι be given.
Assume H3: x4 ∈ x2.
Apply ReplI with
x2,
x1,
x4.
The subproof is completed by applying H3.
Let x4 of type ι be given.
Assume H3: x4 ∈ x2.
Let x5 of type ι be given.
Assume H4: x5 ∈ x2.
Apply H0 with
x4,
x5 leaving 2 subgoals.
Apply H1 with
x4.
The subproof is completed by applying H3.
Apply H1 with
x5.
The subproof is completed by applying H4.
Let x4 of type ι be given.
Assume H3: x4 ∈ {x1 x5|x5 ∈ x2}.
Apply ReplE_impred with
x2,
x1,
x4,
∃ x5 . and (x5 ∈ x2) (x1 x5 = x4) leaving 2 subgoals.
The subproof is completed by applying H3.
Let x5 of type ι be given.
Assume H4: x5 ∈ x2.
Assume H5: x4 = x1 x5.
Let x6 of type ο be given.
Assume H6:
∀ x7 . and (x7 ∈ x2) (x1 x7 = x4) ⟶ x6.
Apply H6 with
x5.
Apply andI with
x5 ∈ x2,
x1 x5 = x4 leaving 2 subgoals.
The subproof is completed by applying H4.
Let x7 of type ι → ι → ο be given.
The subproof is completed by applying H5 with λ x8 x9 . x7 x9 x8.