Nuprl Lemma : presheaf-cumulativity1
∀[C:SmallCategory]. (presheaf{j:l}(C) ⊆r presheaf{[i | j]:l}(C))
Proof
Definitions occuring in Statement :
presheaf: Presheaf(C),
small-category: SmallCategory,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
presheaf: Presheaf(C),
cat-functor: Functor(C1;C2),
type-cat: TypeCat,
all: ∀x:A. B[x],
cat-arrow: cat-arrow(C),
pi1: fst(t),
pi2: snd(t),
cat-id: cat-id(C),
cat-comp: cat-comp(C),
and: P ∧ Q
Lemmas referenced :
small-category_wf,
cat_arrow_triple_lemma,
cat_ob_pair_lemma,
cat-ob_wf,
op-cat_wf,
cat-arrow_wf,
type-cat_wf,
cat-id_wf,
cat-comp_wf,
presheaf_wf1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaEquality_alt,
sqequalRule,
axiomEquality,
hypothesis,
universeIsType,
extract_by_obid,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality_alt,
productElimination,
dependent_pairEquality_alt,
dependent_functionElimination,
Error :memTop,
functionExtensionality,
cumulativity,
applyEquality,
hypothesisEquality,
isectElimination,
functionIsType,
because_Cache,
instantiate,
productIsType,
equalityIstype
Latex:
\mforall{}[C:SmallCategory]. (presheaf\{j:l\}(C) \msubseteq{}r presheaf\{[i | j]:l\}(C))
Date html generated:
2020_05_20-AM-07_52_40
Last ObjectModification:
2020_04_01-AM-00_52_38
Theory : small!categories
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