Nuprl Lemma : presheaf-elements_wf
∀C:SmallCategory. ∀P:Presheaf(C). (el(P) ∈ SmallCategory)
Proof
Definitions occuring in Statement :
presheaf-elements: el(P),
presheaf: Presheaf(C),
small-category: SmallCategory,
all: ∀x:A. B[x],
member: t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
presheaf: Presheaf(C),
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
cat-ob: cat-ob(C),
pi1: fst(t),
type-cat: TypeCat,
cat-arrow: cat-arrow(C),
pi2: snd(t),
prop: ℙ,
presheaf-elements: el(P),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: so_lambda(x,y,z,w,v.t[x; y; z; w; v]),
so_apply: x[s1;s2;s3;s4;s5],
uimplies: b supposing a,
top: Top,
compose: f o g,
and: P ∧ Q,
guard: {T},
squash: ↓T,
true: True
Lemmas referenced :
presheaf_wf,
small-category_wf,
cat-ob_wf,
op-cat_wf,
functor-ob_wf,
small-category-subtype,
type-cat_wf,
subtype_rel_self,
cat-arrow_wf,
equal_wf,
functor-arrow_wf,
mk-cat_wf,
cat-id_wf,
functor-arrow-id,
cat_arrow_triple_lemma,
istype-void,
cat_id_tuple_lemma,
cat-comp_wf,
functor-arrow-comp,
cat_comp_tuple_lemma,
cat-comp-ident,
cat-comp-assoc,
squash_wf,
true_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalHypSubstitution,
hypothesis,
universeIsType,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
productEquality,
applyEquality,
instantiate,
sqequalRule,
universeEquality,
spreadEquality,
setEquality,
because_Cache,
functionEquality,
inhabitedIsType,
productIsType,
productElimination,
setIsType,
equalityIsType1,
lambdaEquality_alt,
independent_isectElimination,
dependent_set_memberEquality_alt,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
applyLambdaEquality,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
independent_pairFormation,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}C:SmallCategory. \mforall{}P:Presheaf(C). (el(P) \mmember{} SmallCategory)
Date html generated:
2019_10_31-AM-07_25_09
Last ObjectModification:
2018_11_10-AM-11_35_10
Theory : small!categories
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