Definition. We define struct_r_equivreln to be λX ⇒ struct_r X ∧ unpack_r_o X (λX' r ⇒ (∀x ∈ X', r x x) ∧ (∀x y ∈ X', r x y → r y x) ∧ (∀x y z ∈ X', r x y → r y z → r x z)) of type set → prop.
In Proofgold the corresponding term root is add478... and object id is 8eed6e...
Theorem. (MetaCat_struct_r_equivreln)
MetaCat struct_r_equivreln Hom_struct_r struct_id struct_comp
In Proofgold the corresponding term root is 9b8747... and proposition id is ca919c...
Proof:
We prove the intermediate claim L1: ∀X, struct_r_equivreln X → struct_r 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_r_gen struct_r_equivreln L1.
∎
Theorem. (MetaCat_struct_r_equivreln_Forgetful)
MetaFunctor struct_r_equivreln Hom_struct_r 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 8e8ae7... and proposition id is a80254...
Proof:
We prove the intermediate claim L1: ∀X, struct_r_equivreln X → struct_r 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_r_Forgetful_gen struct_r_equivreln L1.
∎
Proposition. (MetaCat_struct_r_equivreln_initial)
∃Y : set, ∃uniqa : set → set, initial_p struct_r_equivreln Hom_struct_r struct_id struct_comp Y uniqa
In Proofgold the corresponding term root is 34ae2f... and proposition id is 5dc256...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_equivreln_terminal)
∃Y : set, ∃uniqa : set → set, terminal_p struct_r_equivreln Hom_struct_r struct_id struct_comp Y uniqa
In Proofgold the corresponding term root is ff2837... and proposition id is ae77a0...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_equivreln_coproduct_constr)
∃coprod : set → set → set, ∃i0 i1 : set → set → set, ∃copair : set → set → set → set → set → set, coproduct_constr_p struct_r_equivreln Hom_struct_r struct_id struct_comp coprod i0 i1 copair
In Proofgold the corresponding term root is d99afa... and proposition id is ec184a...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_equivreln_product_constr)
∃prod : set → set → set, ∃pi0 pi1 : set → set → set, ∃pair : set → set → set → set → set → set, product_constr_p struct_r_equivreln Hom_struct_r struct_id struct_comp prod pi0 pi1 pair
In Proofgold the corresponding term root is 78f231... and proposition id is 4d1dff...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_equivreln_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_r_equivreln Hom_struct_r struct_id struct_comp quot canonmap fac
In Proofgold the corresponding term root is 61fb92... and proposition id is 6349ae...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_equivreln_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_r_equivreln Hom_struct_r struct_id struct_comp quot canonmap fac
In Proofgold the corresponding term root is 88bfe6... and proposition id is 99d065...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_equivreln_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_r_equivreln Hom_struct_r struct_id struct_comp po i0 i1 copair
In Proofgold the corresponding term root is 6f21b4... and proposition id is a7d8ad...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_equivreln_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_r_equivreln Hom_struct_r struct_id struct_comp pb pi0 pi1 pair
In Proofgold the corresponding term root is f66e64... and proposition id is e7f7df...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_equivreln_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_r_equivreln Hom_struct_r struct_id struct_comp prod pi0 pi1 pair exp a lm
In Proofgold the corresponding term root is cd35a6... and proposition id is 82496f...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_equivreln_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_r_equivreln Hom_struct_r struct_id struct_comp one uniqa Omega tru ch constr
In Proofgold the corresponding term root is 6853aa... and proposition id is 186505...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_equivreln_nno)
∃one : set, ∃uniqa : set → set, ∃N : set, ∃zer suc : set, ∃rec : set → set → set → set, nno_p struct_r_equivreln Hom_struct_r struct_id struct_comp one uniqa N zer suc rec
In Proofgold the corresponding term root is 86f999... and proposition id is eece97...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_equivreln_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_r_equivreln Hom_struct_r struct_id struct_comp F0 F1 (λX ⇒ X 0) (λX Y f ⇒ f) eta eps
In Proofgold the corresponding term root is d9aa9b... and proposition id is 80d3dc...
Proof:
The rest of the proof is missing.