Nuprl Lemma : presheaf-fun-family-discrete
∀[C:SmallCategory]. ∀[A,B:Type]. ∀[X:ps_context{j:l}(C)]. ∀[I:cat-ob(C)]. ∀[a:X(I)].
(presheaf-fun-family(C; X; discr(A); discr(B); I; a)
= {w:J:cat-ob(C) ⟶ f:(cat-arrow(C) J I) ⟶ u:A ⟶ B|
∀J,K:cat-ob(C). ∀f:cat-arrow(C) J I. ∀g:cat-arrow(C) K J. ∀u:A.
((w J f u) = (w K (cat-comp(C) K J I g f) u) ∈ B)}
∈ Type)
Proof
Definitions occuring in Statement :
discrete-presheaf-type: discr(T),
presheaf-fun-family: presheaf-fun-family(C; X; A; B; I; a),
I_set: A(I),
ps_context: __⊢,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
cat-comp: cat-comp(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
discrete-presheaf-type: discr(T),
presheaf-fun-family: presheaf-fun-family(C; X; A; B; I; a),
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B
Lemmas referenced :
presheaf_type_at_pair_lemma,
presheaf_type_ap_morph_pair_lemma,
cat-ob_wf,
cat-arrow_wf,
equal_wf,
cat-comp_wf,
I_set_wf,
ps_context_wf,
small-category-cumulativity-2,
istype-universe,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
Error :memTop,
hypothesis,
setEquality,
functionEquality,
isectElimination,
hypothesisEquality,
applyEquality,
universeIsType,
instantiate,
because_Cache,
universeEquality
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[A,B:Type]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[I:cat-ob(C)]. \mforall{}[a:X(I)].
(presheaf-fun-family(C; X; discr(A); discr(B); I; a)
= \{w:J:cat-ob(C) {}\mrightarrow{} f:(cat-arrow(C) J I) {}\mrightarrow{} u:A {}\mrightarrow{} B|
\mforall{}J,K:cat-ob(C). \mforall{}f:cat-arrow(C) J I. \mforall{}g:cat-arrow(C) K J. \mforall{}u:A.
((w J f u) = (w K (cat-comp(C) K J I g f) u))\} )
Date html generated:
2020_05_20-PM-01_35_43
Last ObjectModification:
2020_04_02-PM-06_35_46
Theory : presheaf!models!of!type!theory
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