Proofgold Address

Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Definition bij^bij := λ x0 x1 . λ x2 : ι → ι . and (and (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x1) (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4)) (∀ x3 . x3 ∈ x1 ⟶ ∀ x4 : ο . (∀ x5 . and (x5 ∈ x0) (x2 x5 = x3) ⟶ x4) ⟶ x4)
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Theorem bijI^bijI : ∀ x0 x1 . ∀ x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x1) ⟶ (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4) ⟶ (∀ x3 . x3 ∈ x1 ⟶ ∀ x4 : ο . (∀ x5 . and (x5 ∈ x0) (x2 x5 = x3) ⟶ x4) ⟶ x4) ⟶ bij x0 x1 x2
Theorem bijE^bijE : ∀ x0 x1 . ∀ x2 : ι → ι . bij x0 x1 x2 ⟶ ∀ x3 : ο . ((∀ x4 . x4 ∈ x0 ⟶ x2 x4 ∈ x1) ⟶ (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x2 x4 = x2 x5 ⟶ x4 = x5) ⟶ (∀ x4 . x4 ∈ x1 ⟶ ∀ x5 : ο . (∀ x6 . and (x6 ∈ x0) (x2 x6 = x4) ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3
Definition equip^equip := λ x0 x1 . ∀ x2 : ο . (∀ x3 : ι → ι . bij x0 x1 x3 ⟶ x2) ⟶ x2
Known bij_id^bij_id : ∀ x0 . bij x0 x0 (λ x1 . x1)
Theorem equip_ref^equip_ref : ∀ x0 . equip x0 x0
Param inv^inv : ι → (ι → ι) → ι → ι
Known bij_inv^bij_inv : ∀ x0 x1 . ∀ x2 : ι → ι . bij x0 x1 x2 ⟶ bij x1 x0 (inv x0 x2)
Theorem equip_sym^equip_sym : ∀ x0 x1 . equip x0 x1 ⟶ equip x1 x0
Known bij_comp^bij_comp : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι . bij x0 x1 x3 ⟶ bij x1 x2 x4 ⟶ bij x0 x2 (λ x5 . x4 (x3 x5))
Theorem equip_tra^equip_tra : ∀ x0 x1 x2 . equip x0 x1 ⟶ equip x1 x2 ⟶ equip x0 x2