Nuprl Lemma : presheaf-type-iso-inverse
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)].
((presheaf-type-rev-iso(X) o presheaf-type-iso(X)) = (λx.x) ∈ (presheaf_type{i:l}(C; X) ⟶ presheaf_type{i:l}(C; X)))
Proof
Definitions occuring in Statement :
presheaf-type-rev-iso: presheaf-type-rev-iso(X),
presheaf-type-iso: presheaf-type-iso(X),
presheaf_type: presheaf_type{i:l}(C; X),
ps_context: __⊢,
compose: f o g,
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
function: x:A ⟶ B[x],
equal: s = t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
compose: f o g,
presheaf_type: presheaf_type{i:l}(C; X),
presheaf: Presheaf(C),
cat-functor: Functor(C1;C2),
subtype_rel: A ⊆r B,
and: P ∧ Q,
presheaf-type-iso: presheaf-type-iso(X),
presheaf-type-rev-iso: presheaf-type-rev-iso(X),
pi1: fst(t),
all: ∀x:A. B[x],
pi2: snd(t),
mk-presheaf: mk-presheaf,
mk-functor: mk-functor,
ps_context: __⊢,
uimplies: b supposing a,
cat-ob: cat-ob(C),
type-cat: TypeCat,
functor-ob: ob(F),
I_set: A(I),
prop: ℙ
Lemmas referenced :
presheaf_type_wf,
ps_context_cumulativity2,
small-category-cumulativity-2,
ps_context_wf,
small-category_wf,
sets_wf,
sets-ob,
sets-arrow,
sets-comp,
sets-id,
ob_pair_lemma,
arrow_pair_lemma,
cat-ob_wf,
I_set_wf,
cat-arrow_wf,
functor-ob_wf,
op-cat_wf,
type-cat_wf,
subtype_rel-equal,
cat_ob_op_lemma,
subtype_rel_self,
psc-restriction_wf,
cat-id_wf,
cat-comp_wf,
op-cat-id,
op-cat-arrow,
op-cat-comp,
cat_arrow_triple_lemma,
cat_comp_tuple_lemma,
cat_id_tuple_lemma,
cat_ob_pair_lemma,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
functionExtensionality,
sqequalRule,
equalitySymmetry,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
dependent_set_memberEquality_alt,
hypothesis,
instantiate,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
universeIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
because_Cache,
Error :memTop,
dependent_functionElimination,
dependent_pairEquality_alt,
productEquality,
cumulativity,
functionIsType,
productIsType,
setIsType,
equalityIstype,
independent_isectElimination,
universeEquality,
lambdaEquality_alt,
equalityTransitivity,
setEquality
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)].
((presheaf-type-rev-iso(X) o presheaf-type-iso(X)) = (\mlambda{}x.x))
Date html generated:
2020_05_20-PM-01_25_40
Last ObjectModification:
2020_04_02-AM-11_05_08
Theory : presheaf!models!of!type!theory
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