Let x0 of type ι be given.

Let x1 of type ο be given.

Assume H0: ∀ x2 : ι → ι . bij (Sing x0) u1 x2 ⟶ x1.

Apply H0 with λ x2 . 0.

Apply bijI with Sing x0, u1, λ x2 . 0 leaving 3 subgoals.

Let x2 of type ι be given.

Assume H1: x2 ∈ Sing x0.

The subproof is completed by applying In_0_1.

Let x2 of type ι be given.

Assume H1: x2 ∈ Sing x0.

Let x3 of type ι be given.

Assume H2: x3 ∈ Sing x0.

Assume H3: (λ x4 . 0) x2 = (λ x4 . 0) x3.

Apply SingE with x0, x3, λ x4 x5 . x2 = x5 leaving 2 subgoals.

The subproof is completed by applying H2.

Apply SingE with x0, x2.

The subproof is completed by applying H1.

Let x2 of type ι be given.

Assume H1: x2 ∈ u1.

Apply cases_1 with x2, λ x3 . ∃ x4 . and (x4 ∈ Sing x0) ((λ x5 . 0) x4 = x3) leaving 2 subgoals.

The subproof is completed by applying H1.

Let x3 of type ο be given.

Assume H2: ∀ x4 . and (x4 ∈ Sing x0) ((λ x5 . 0) x4 = 0) ⟶ x3.

Apply H2 with x0.

Apply andI with x0 ∈ Sing x0, (λ x4 . 0) x0 = 0 leaving 2 subgoals.

The subproof is completed by applying SingI with x0.

Let x4 of type ι → ι → ο be given.

Assume H3: x4 ((λ x5 . 0) x0) 0.

The subproof is completed by applying H3.

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Proofgold Proof