Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι be given.
Assume H0:
x2 ∈ lam x0 (λ x3 . x1 x3).
Claim L1:
∃ x3 . and (x3 ∈ x0) (x2 ∈ {setsum x3 x4|x4 ∈ x1 x3})Apply famunionE with
x0,
λ x3 . {setsum x3 x4|x4 ∈ x1 x3},
x2.
The subproof is completed by applying H0.
Apply exandE_i with
λ x3 . x3 ∈ x0,
λ x3 . x2 ∈ {setsum x3 x4|x4 ∈ x1 x3},
and (and (setsum (proj0 x2) (proj1 x2) = x2) (proj0 x2 ∈ x0)) (proj1 x2 ∈ x1 (proj0 x2)) leaving 2 subgoals.
The subproof is completed by applying L1.
Let x3 of type ι be given.
Assume H2: x3 ∈ x0.
Assume H3:
x2 ∈ {setsum x3 x4|x4 ∈ x1 x3}.
Apply ReplE_impred with
x1 x3,
setsum x3,
x2,
and (and (setsum (proj0 x2) (proj1 x2) = x2) (proj0 x2 ∈ x0)) (proj1 x2 ∈ x1 (proj0 x2)) leaving 2 subgoals.
The subproof is completed by applying H3.
Let x4 of type ι be given.
Assume H4: x4 ∈ x1 x3.
Apply H5 with
λ x5 x6 . and (and (setsum (proj0 x6) (proj1 x6) = x6) (proj0 x6 ∈ x0)) (proj1 x6 ∈ x1 (proj0 x6)).
Apply proj0_pair_eq with
x3,
x4,
λ x5 x6 . and (and (setsum x6 (proj1 (setsum x3 x4)) = setsum x3 x4) (x6 ∈ x0)) (proj1 (setsum x3 x4) ∈ x1 x6).
Apply proj1_pair_eq with
x3,
x4,
λ x5 x6 . and (and (setsum x3 x6 = setsum x3 x4) (x3 ∈ x0)) (x6 ∈ x1 x3).
Apply and3I with
setsum x3 x4 = setsum x3 x4,
x3 ∈ x0,
x4 ∈ x1 x3 leaving 3 subgoals.
Let x5 of type ι → ι → ο be given.
The subproof is completed by applying H6.
The subproof is completed by applying H2.
The subproof is completed by applying H4.