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 x3 = x2 x3.
Let x3 of type ι be given.
Assume H1:
x3 ∈ lam x0 (λ x4 . x1 x4).
Claim L2:
∃ x4 . and (x4 ∈ x0) (∃ x5 . and (x5 ∈ x1 x4) (x3 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5)))Apply lamE2 with
x0,
x1,
x3.
The subproof is completed by applying H1.
Apply L2 with
x3 ∈ lam x0 x2.
Let x4 of type ι be given.
Assume H3:
(λ x5 . and (x5 ∈ x0) (∃ x6 . and (x6 ∈ x1 x5) (x3 = lam 2 (λ x7 . If_i (x7 = 0) x5 x6)))) x4.
Apply H3 with
x3 ∈ lam x0 x2.
Assume H4: x4 ∈ x0.
Assume H5:
∃ x5 . and (x5 ∈ x1 x4) (x3 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5)).
Apply H5 with
x3 ∈ lam x0 x2.
Let x5 of type ι be given.
Assume H6:
(λ x6 . and (x6 ∈ x1 x4) (x3 = lam 2 (λ x7 . If_i (x7 = 0) x4 x6))) x5.
Apply H6 with
x3 ∈ lam x0 x2.
Assume H7: x5 ∈ x1 x4.
Assume H8:
x3 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5).
Apply H8 with
λ x6 x7 . x7 ∈ lam x0 (λ x8 . x2 x8).
Apply tuple_2_Sigma with
x0,
λ x6 . x2 x6,
x4,
x5 leaving 2 subgoals.
The subproof is completed by applying H4.
Apply H0 with
x4,
λ x6 x7 . x5 ∈ x6 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H7.