Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Apply set_ext with
{x1 x2|x2 ∈ x0},
famunion x0 (λ x2 . Sing (x1 x2)) leaving 2 subgoals.
Let x2 of type ι be given.
Assume H0: x2 ∈ {x1 x3|x3 ∈ x0}.
Apply ReplE_impred with
x0,
x1,
x2,
x2 ∈ famunion x0 (λ x3 . Sing (x1 x3)) leaving 2 subgoals.
The subproof is completed by applying H0.
Let x3 of type ι be given.
Assume H1: x3 ∈ x0.
Assume H2: x2 = x1 x3.
Apply famunionI with
x0,
λ x4 . Sing (x1 x4),
x3,
x2 leaving 2 subgoals.
The subproof is completed by applying H1.
Apply H2 with
λ x4 x5 . x2 ∈ Sing x4.
The subproof is completed by applying SingI with x2.
Let x2 of type ι be given.
Apply famunionE with
x0,
λ x3 . Sing (x1 x3),
x2,
x2 ∈ {x1 x3|x3 ∈ x0} leaving 2 subgoals.
The subproof is completed by applying H0.
Let x3 of type ι be given.
Assume H1:
and (x3 ∈ x0) (x2 ∈ Sing (x1 x3)).
Apply H1 with
x2 ∈ {x1 x4|x4 ∈ x0}.
Assume H2: x3 ∈ x0.
Assume H3:
x2 ∈ Sing (x1 x3).
Claim L4: x2 = x1 x3
Apply SingE with
x1 x3,
x2.
The subproof is completed by applying H3.
Apply L4 with
λ x4 x5 . x5 ∈ {x1 x6|x6 ∈ x0}.
Apply ReplI with
x0,
x1,
x3.
The subproof is completed by applying H2.