Nuprl Lemma : rep-pre-sheaf_wf
∀[C:SmallCategory]. ∀[X:cat-ob(C)]. (rep-pre-sheaf(C;X) ∈ Functor(op-cat(C);TypeCat))
Proof
Definitions occuring in Statement :
rep-pre-sheaf: rep-pre-sheaf(C;X),
type-cat: TypeCat,
op-cat: op-cat(C),
cat-functor: Functor(C1;C2),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
cand: A c∧ B,
and: P ∧ Q,
compose: f o g,
cat-id: cat-id(C),
subtype_rel: A ⊆r B,
pi2: snd(t),
pi1: fst(t),
spreadn: spread4,
cat-ob: cat-ob(C),
op-cat: op-cat(C),
type-cat: TypeCat,
cat-arrow: cat-arrow(C),
cat-comp: cat-comp(C),
small-category: SmallCategory,
cat-functor: Functor(C1;C2),
rep-pre-sheaf: rep-pre-sheaf(C;X),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
small-category_wf,
cat-comp_wf,
cat-id_wf,
type-cat_wf,
cat-arrow_wf,
equal_wf,
op-cat_wf,
cat-ob_wf,
all_wf,
subtype_rel_self
Rules used in proof :
isect_memberEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
universeEquality,
instantiate,
productEquality,
independent_pairFormation,
dependent_functionElimination,
functionExtensionality,
lambdaFormation,
functionEquality,
because_Cache,
cumulativity,
hypothesis,
isectElimination,
lemma_by_obid,
hypothesisEquality,
applyEquality,
lambdaEquality,
dependent_pairEquality,
sqequalRule,
productElimination,
rename,
thin,
setElimination,
sqequalHypSubstitution,
dependent_set_memberEquality,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:cat-ob(C)]. (rep-pre-sheaf(C;X) \mmember{} Functor(op-cat(C);TypeCat))
Date html generated:
2020_05_20-AM-07_53_01
Last ObjectModification:
2015_12_28-PM-02_23_30
Theory : small!categories
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