Nuprl Lemma : ps-discrete-map-is-constant
∀[C:SmallCategory]. ∀[T:𝕌{j}]. ∀[I:cat-ob(C)]. ∀[s:psc_map{[i | j]:l}(C; Yoneda(I); discrete-set(T))].
(s = (λJ,g. (s I (cat-id(C) I))) ∈ psc_map{[i | j]:l}(C; Yoneda(I); discrete-set(T)))
Proof
Definitions occuring in Statement :
psc_map: A ⟶ B,
discrete-set: discrete-set(A),
Yoneda: Yoneda(I),
uall: ∀[x:A]. B[x],
apply: f a,
lambda: λx.A[x],
universe: Type,
equal: s = t ∈ T,
cat-id: cat-id(C),
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
Yoneda: Yoneda(I),
discrete-set: discrete-set(A),
type-cat: TypeCat,
all: ∀x:A. B[x],
member: t ∈ T,
compose: f o g,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
and: P ∧ Q
Lemmas referenced :
cat_arrow_triple_lemma,
ob_pair_lemma,
cat_comp_tuple_lemma,
arrow_pair_lemma,
psc_map_wf,
small-category-cumulativity-2,
Yoneda_wf,
discrete-set_wf,
cat-ob_wf,
istype-universe,
small-category_wf,
subtype_rel-equal,
op-cat_wf,
cat_ob_op_lemma,
cat-arrow_wf,
op-cat-arrow,
cat-id_wf,
cat-comp-ident,
cat-comp_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
sqequalRule,
introduction,
extract_by_obid,
dependent_functionElimination,
Error :memTop,
hypothesis,
dependent_set_memberEquality_alt,
universeIsType,
instantiate,
isectElimination,
hypothesisEquality,
applyEquality,
cumulativity,
universeEquality,
functionExtensionality,
because_Cache,
independent_isectElimination,
applyLambdaEquality,
productElimination,
equalityTransitivity,
equalitySymmetry,
functionIsType,
equalityIstype,
lambdaEquality_alt
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[T:\mBbbU{}\{j\}]. \mforall{}[I:cat-ob(C)].
\mforall{}[s:psc\_map\{[i | j]:l\}(C; Yoneda(I); discrete-set(T))].
(s = (\mlambda{}J,g. (s I (cat-id(C) I))))
Date html generated:
2020_05_20-PM-01_34_33
Last ObjectModification:
2020_04_03-PM-00_49_47
Theory : presheaf!models!of!type!theory
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