Definition. We define struct_c_topology to be λX ⇒ struct_c X ∧ unpack_c_o X (λX' Open ⇒ Open (λx ⇒ x ∈ X') ∧ (∀U V : set → prop, Open U → Open V → Open (λx ⇒ U x ∧ U x)) ∧ (∀C : (set → prop) → prop, (∀U : set → prop, C U → Open U) → Open (λx ⇒ ∃U : set → prop, C U ∧ U x))) of type set → prop.
In Proofgold the corresponding term root is 7943e0... and object id is 90dfac...
Theorem. (MetaCat_struct_c_topology)
MetaCat struct_c_topology Hom_struct_c struct_id struct_comp
In Proofgold the corresponding term root is 113fe0... and proposition id is c55834...
Proof:
We prove the intermediate claim L1: ∀X, struct_c_topology X → struct_c X.
Let X be given.
Assume HX.
Apply HX to the current goal.
Assume H _.
An exact proof term for the current goal is H.
An exact proof term for the current goal is MetaCat_struct_c_gen struct_c_topology L1.
∎
Theorem. (MetaCat_struct_c_topology_Forgetful)
MetaFunctor struct_c_topology Hom_struct_c struct_id struct_comp (λ_ ⇒ True) SetHom (λX ⇒ lam_id X) (λX Y Z f g ⇒ (lam_comp X f g)) (λX ⇒ X 0) (λX Y f ⇒ f)
In Proofgold the corresponding term root is d11d37... and proposition id is 8c0bb7...
Proof:
We prove the intermediate claim L1: ∀X, struct_c_topology X → struct_c X.
Let X be given.
Assume HX.
Apply HX to the current goal.
Assume H _.
An exact proof term for the current goal is H.
An exact proof term for the current goal is MetaCat_struct_c_Forgetful_gen struct_c_topology L1.
∎
Proposition. (MetaCat_struct_c_topology_initial)
∃Y : set, ∃uniqa : set → set, initial_p struct_c_topology Hom_struct_c struct_id struct_comp Y uniqa
In Proofgold the corresponding term root is 0494e1... and proposition id is 820104...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_c_topology_terminal)
∃Y : set, ∃uniqa : set → set, terminal_p struct_c_topology Hom_struct_c struct_id struct_comp Y uniqa
In Proofgold the corresponding term root is 691b34... and proposition id is 92cff0...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_c_topology_coproduct_constr)
∃coprod : set → set → set, ∃i0 i1 : set → set → set, ∃copair : set → set → set → set → set → set, coproduct_constr_p struct_c_topology Hom_struct_c struct_id struct_comp coprod i0 i1 copair
In Proofgold the corresponding term root is c07b24... and proposition id is 138906...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_c_topology_product_constr)
∃prod : set → set → set, ∃pi0 pi1 : set → set → set, ∃pair : set → set → set → set → set → set, product_constr_p struct_c_topology Hom_struct_c struct_id struct_comp prod pi0 pi1 pair
In Proofgold the corresponding term root is 796d4d... and proposition id is e8f2e9...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_c_topology_coequalizer_constr)
∃quot : set → set → set → set → set, ∃canonmap : set → set → set → set → set, ∃fac : set → set → set → set → set → set → set, coequalizer_constr_p struct_c_topology Hom_struct_c struct_id struct_comp quot canonmap fac
In Proofgold the corresponding term root is 9152c2... and proposition id is 050361...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_c_topology_equalizer_constr)
∃quot : set → set → set → set → set, ∃canonmap : set → set → set → set → set, ∃fac : set → set → set → set → set → set → set, equalizer_constr_p struct_c_topology Hom_struct_c struct_id struct_comp quot canonmap fac
In Proofgold the corresponding term root is 30bbf3... and proposition id is ae2fd3...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_c_topology_pushout_constr)
∃po : set → set → set → set → set → set, ∃i0 : set → set → set → set → set → set, ∃i1 : set → set → set → set → set → set, ∃copair : set → set → set → set → set → set → set → set → set, pushout_constr_p struct_c_topology Hom_struct_c struct_id struct_comp po i0 i1 copair
In Proofgold the corresponding term root is fa2232... and proposition id is d54c49...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_c_topology_pullback_constr)
∃pb : set → set → set → set → set → set, ∃pi0 : set → set → set → set → set → set, ∃pi1 : set → set → set → set → set → set, ∃pair : set → set → set → set → set → set → set → set → set, pullback_constr_p struct_c_topology Hom_struct_c struct_id struct_comp pb pi0 pi1 pair
In Proofgold the corresponding term root is 72db7b... and proposition id is ebec36...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_c_topology_product_exponent)
∃prod : set → set → set, ∃pi0 pi1 : set → set → set, ∃pair : set → set → set → set → set → set, ∃exp : set → set → set, ∃a : set → set → set, ∃lm : set → set → set → set → set, product_exponent_constr_p struct_c_topology Hom_struct_c struct_id struct_comp prod pi0 pi1 pair exp a lm
In Proofgold the corresponding term root is 78a433... and proposition id is a69bb4...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_c_topology_subobject_classifier)
∃one : set, ∃uniqa : set → set, ∃Omega : set, ∃tru : set, ∃ch : set → set → set → set, ∃constr : set → set → set → set → set → set → set, subobject_classifier_p struct_c_topology Hom_struct_c struct_id struct_comp one uniqa Omega tru ch constr
In Proofgold the corresponding term root is 0f6b6b... and proposition id is 1a6d49...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_c_topology_nno)
∃one : set, ∃uniqa : set → set, ∃N : set, ∃zer suc : set, ∃rec : set → set → set → set, nno_p struct_c_topology Hom_struct_c struct_id struct_comp one uniqa N zer suc rec
In Proofgold the corresponding term root is ec504a... and proposition id is b6cb9a...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_c_topology_left_adjoint_forgetful)
∃F0 : set → set, ∃F1 : set → set → set → set, ∃eta eps : set → set, MetaAdjunction_strict (λ_ ⇒ True) SetHom (λX ⇒ (lam_id X)) (λX Y Z f g ⇒ (lam_comp X f g)) struct_c_topology Hom_struct_c struct_id struct_comp F0 F1 (λX ⇒ X 0) (λX Y f ⇒ f) eta eps
In Proofgold the corresponding term root is 0b89b5... and proposition id is 59fb45...
Proof:
The rest of the proof is missing.