Definition. We define struct_u_bij to be λX ⇒ struct_u X ∧ unpack_u_o X (λX' h ⇒ bij X' X' (λx ⇒ h x)) of type set → prop.
In Proofgold the corresponding term root is 4f4ed8... and object id is e29a29...
Theorem. (MetaCat_struct_u_bij)
MetaCat struct_u_bij Hom_struct_u struct_id struct_comp
In Proofgold the corresponding term root is 97a117... and proposition id is 17fce1...
Proof:
We prove the intermediate claim L1: ∀X, struct_u_bij X → struct_u 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_u_gen struct_u_bij L1.
∎
Theorem. (MetaCat_struct_u_bij_Forgetful)
MetaFunctor struct_u_bij Hom_struct_u 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 35386a... and proposition id is ac3641...
Proof:
We prove the intermediate claim L1: ∀X, struct_u_bij X → struct_u 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_u_Forgetful_gen struct_u_bij L1.
∎
Proposition. (MetaCat_struct_u_bij_initial)
∃Y : set, ∃uniqa : set → set, initial_p struct_u_bij Hom_struct_u struct_id struct_comp Y uniqa
In Proofgold the corresponding term root is 54ae7b... and proposition id is 955bd3...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_u_bij_terminal)
∃Y : set, ∃uniqa : set → set, terminal_p struct_u_bij Hom_struct_u struct_id struct_comp Y uniqa
In Proofgold the corresponding term root is aed94d... and proposition id is 32041b...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_u_bij_coproduct_constr)
∃coprod : set → set → set, ∃i0 i1 : set → set → set, ∃copair : set → set → set → set → set → set, coproduct_constr_p struct_u_bij Hom_struct_u struct_id struct_comp coprod i0 i1 copair
In Proofgold the corresponding term root is c54945... and proposition id is fc15b3...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_u_bij_product_constr)
∃prod : set → set → set, ∃pi0 pi1 : set → set → set, ∃pair : set → set → set → set → set → set, product_constr_p struct_u_bij Hom_struct_u struct_id struct_comp prod pi0 pi1 pair
In Proofgold the corresponding term root is 6b82cc... and proposition id is 6ce535...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_u_bij_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_u_bij Hom_struct_u struct_id struct_comp quot canonmap fac
In Proofgold the corresponding term root is bb5efb... and proposition id is b2874a...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_u_bij_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_u_bij Hom_struct_u struct_id struct_comp quot canonmap fac
In Proofgold the corresponding term root is 1c2881... and proposition id is 246041...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_u_bij_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_u_bij Hom_struct_u struct_id struct_comp po i0 i1 copair
In Proofgold the corresponding term root is f5a515... and proposition id is 5fc899...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_u_bij_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_u_bij Hom_struct_u struct_id struct_comp pb pi0 pi1 pair
In Proofgold the corresponding term root is cdc686... and proposition id is 6f6f35...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_u_bij_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_u_bij Hom_struct_u struct_id struct_comp prod pi0 pi1 pair exp a lm
In Proofgold the corresponding term root is 8586e3... and proposition id is c56107...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_u_bij_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_u_bij Hom_struct_u struct_id struct_comp one uniqa Omega tru ch constr
In Proofgold the corresponding term root is c685d4... and proposition id is a1bf45...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_u_bij_nno)
∃one : set, ∃uniqa : set → set, ∃N : set, ∃zer suc : set, ∃rec : set → set → set → set, nno_p struct_u_bij Hom_struct_u struct_id struct_comp one uniqa N zer suc rec
In Proofgold the corresponding term root is aab681... and proposition id is cdd84d...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_u_bij_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_u_bij Hom_struct_u struct_id struct_comp F0 F1 (λX ⇒ X 0) (λX Y f ⇒ f) eta eps
In Proofgold the corresponding term root is 062a21... and proposition id is a69df3...
Proof:
The rest of the proof is missing.