Nuprl Lemma : presheaf-element-map_wf
∀[C:SmallCategory]. ∀[A,B:Presheaf(C)]. ∀[m:A ⟶ B]. (presheaf-element-map(m) ∈ Functor(el(A);el(B)))
Proof
Definitions occuring in Statement :
presheaf-element-map: presheaf-element-map(m),
presheaf_map: A ⟶ B,
presheaf-elements: el(P),
presheaf: Presheaf(C),
cat-functor: Functor(C1;C2),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
presheaf-element-map: presheaf-element-map(m),
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
uimplies: b supposing a,
presheaf-elements: el(P),
mk-cat: mk-cat,
top: Top,
presheaf_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
subtype_rel: A ⊆r B,
cat-arrow: cat-arrow(C),
pi1: fst(t),
pi2: snd(t),
type-cat: TypeCat,
presheaf: Presheaf(C),
cat-ob: cat-ob(C),
prop: ℙ,
compose: f o g
Lemmas referenced :
mk-functor_wf,
presheaf-elements_wf,
presheaf_map_wf,
presheaf_wf,
small-category_wf,
cat_ob_pair_lemma,
subtype_rel_self,
functor-ob_wf,
op-cat_wf,
type-cat_wf,
cat-ob_wf,
small-category-subtype,
cat_arrow_triple_lemma,
cat-arrow_wf,
equal_wf,
functor-arrow_wf,
cat_comp_tuple_lemma,
cat-comp_wf,
functor-arrow-comp,
cat_id_tuple_lemma,
cat-id_wf,
functor-arrow-id
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
sqequalRule,
independent_isectElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
because_Cache,
voidElimination,
voidEquality,
functionExtensionality,
productElimination,
dependent_pairEquality,
applyEquality,
setElimination,
rename,
functionEquality,
instantiate,
productEquality,
universeEquality,
dependent_set_memberEquality,
applyLambdaEquality,
lambdaFormation
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[A,B:Presheaf(C)]. \mforall{}[m:A {}\mrightarrow{} B].
(presheaf-element-map(m) \mmember{} Functor(el(A);el(B)))
Date html generated:
2017_10_05-AM-00_51_29
Last ObjectModification:
2017_10_04-PM-06_55_47
Theory : small!categories
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