Definition. We define struct_u_idem to be λX ⇒ struct_u X ∧ unpack_u_o X (λX' h ⇒ ∀x ∈ X', h (h x) = h x) of type set → prop.
In Proofgold the corresponding term root is a66564... and object id is 9f2532...
Theorem. (MetaCat_struct_u_idem)
MetaCat struct_u_idem Hom_struct_u struct_id struct_comp
In Proofgold the corresponding term root is 2568db... and proposition id is 31a6e5...
Proof:
We prove the intermediate claim L1: ∀X, struct_u_idem 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_idem L1.
∎
Theorem. (MetaCat_struct_u_idem_Forgetful)
MetaFunctor struct_u_idem 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 f09d41... and proposition id is 3605e1...
Proof:
We prove the intermediate claim L1: ∀X, struct_u_idem 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_idem L1.
∎
Proposition. (MetaCat_struct_u_idem_initial)
∃Y : set, ∃uniqa : set → set, initial_p struct_u_idem Hom_struct_u struct_id struct_comp Y uniqa
In Proofgold the corresponding term root is 6535c5... and proposition id is 2fd91b...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_u_idem_terminal)
∃Y : set, ∃uniqa : set → set, terminal_p struct_u_idem Hom_struct_u struct_id struct_comp Y uniqa
In Proofgold the corresponding term root is 6f1f95... and proposition id is cae128...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_u_idem_coproduct_constr)
∃coprod : set → set → set, ∃i0 i1 : set → set → set, ∃copair : set → set → set → set → set → set, coproduct_constr_p struct_u_idem Hom_struct_u struct_id struct_comp coprod i0 i1 copair
In Proofgold the corresponding term root is 23071a... and proposition id is 18e31e...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_u_idem_product_constr)
∃prod : set → set → set, ∃pi0 pi1 : set → set → set, ∃pair : set → set → set → set → set → set, product_constr_p struct_u_idem Hom_struct_u struct_id struct_comp prod pi0 pi1 pair
In Proofgold the corresponding term root is 4dd455... and proposition id is f2a7fc...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_u_idem_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_idem Hom_struct_u struct_id struct_comp quot canonmap fac
In Proofgold the corresponding term root is 3a3d04... and proposition id is fdf827...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_u_idem_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_idem Hom_struct_u struct_id struct_comp quot canonmap fac
In Proofgold the corresponding term root is b2b78f... and proposition id is 877ff5...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_u_idem_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_idem Hom_struct_u struct_id struct_comp po i0 i1 copair
In Proofgold the corresponding term root is 4dcd45... and proposition id is 685b03...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_u_idem_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_idem Hom_struct_u struct_id struct_comp pb pi0 pi1 pair
In Proofgold the corresponding term root is b10396... and proposition id is 5b67a3...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_u_idem_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_idem Hom_struct_u struct_id struct_comp prod pi0 pi1 pair exp a lm
In Proofgold the corresponding term root is ef6ef4... and proposition id is 2b51e7...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_u_idem_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_idem Hom_struct_u struct_id struct_comp one uniqa Omega tru ch constr
In Proofgold the corresponding term root is 1364e8... and proposition id is 7c0d04...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_u_idem_nno)
∃one : set, ∃uniqa : set → set, ∃N : set, ∃zer suc : set, ∃rec : set → set → set → set, nno_p struct_u_idem Hom_struct_u struct_id struct_comp one uniqa N zer suc rec
In Proofgold the corresponding term root is 61fe8e... and proposition id is 8d7528...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_u_idem_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_idem 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 533ea1... and proposition id is 49304d...
Proof:
The rest of the proof is missing.