Apply unknownprop_795e291855a044d4d636c961caa1600704603cc02e33a7b37aa66e8d7f6512db with
struct_e,
PtdSetHom,
λ x0 . lam_id (ap x0 0),
λ x0 x1 x2 x3 x4 . lam_comp (ap x0 0) x3 x4,
PtdPred,
UnaryPredHom,
λ x0 . lam_id (ap x0 0),
λ x0 x1 x2 x3 x4 . lam_comp (ap x0 0) x3 x4,
λ x0 . unpack_e_i x0 (λ x1 x2 . pack_p x1 (λ x3 . x3 = x2)),
λ x0 x1 x2 . x2 leaving 4 subgoals.
Let x0 of type ι be given.
Apply H0 with
λ x1 . PtdPred ((λ x2 . unpack_e_i x2 (λ x3 x4 . pack_p x3 (λ x5 . x5 = x4))) x1).
Let x1 of type ι be given.
Let x2 of type ι be given.
Assume H1: x2 ∈ x1.
Apply unpack_e_i_eq with
λ x3 x4 . pack_p x3 (λ x5 . x5 = x4),
x1,
x2,
λ x3 x4 . PtdPred x4.
Apply unknownprop_2576d2815b46016e5e13a9989b4e9789629d83c56ed1c92a4cda0de077a20752 with
x1,
λ x3 . x3 = x2.
Let x3 of type ο be given.
Assume H2:
∀ x4 . and (x4 ∈ x1) ((λ x5 . x5 = x2) x4) ⟶ x3.
Apply H2 with
x2.
Apply andI with
x2 ∈ x1,
(λ x4 . x4 = x2) x2 leaving 2 subgoals.
The subproof is completed by applying H1.
Let x4 of type ι → ι → ο be given.
Assume H3: x4 x2 x2.
The subproof is completed by applying H3.
Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply H0 with
λ x3 . PtdSetHom x3 x1 x2 ⟶ UnaryPredHom ((λ x4 . unpack_e_i x4 (λ x5 x6 . pack_p x5 (λ x7 . x7 = x6))) x3) ((λ x4 . unpack_e_i x4 (λ x5 x6 . pack_p x5 (λ x7 . x7 = x6))) x1) ((λ x4 x5 x6 . x6) x3 x1 x2).
Let x3 of type ι be given.
Let x4 of type ι be given.
Assume H2: x4 ∈ x3.
Apply H1 with
λ x5 . PtdSetHom (pack_e x3 x4) x5 x2 ⟶ UnaryPredHom ((λ x6 . unpack_e_i x6 (λ x7 x8 . pack_p x7 (λ x9 . x9 = x8))) (pack_e x3 x4)) ((λ x6 . unpack_e_i x6 (λ x7 x8 . pack_p x7 (λ x9 . x9 = x8))) x5) ((λ x6 x7 x8 . x8) (pack_e x3 x4) x5 x2).
Let x5 of type ι be given.
Let x6 of type ι be given.
Assume H3: x6 ∈ x5.
Apply unknownprop_266cf3934e79ff708b43f6101066886a004a8b2cb57b38750ae943dbc174c7c9 with
x3,
x5,
x4,
x6,
x2,
λ x7 x8 : ο . x8 ⟶ UnaryPredHom ((λ x9 . unpack_e_i x9 (λ x10 x11 . pack_p x10 (λ x12 . x12 = x11))) (pack_e x3 x4)) ((λ x9 . unpack_e_i x9 (λ x10 x11 . pack_p x10 (λ x12 . x12 = x11))) (pack_e x5 x6)) ((λ x9 x10 x11 . x11) (pack_e x3 x4) (pack_e x5 x6) x2).