Nuprl Lemma : presheaf-term-at-morph
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[u:{X ⊢ _:A}]. ∀[I:cat-ob(C)]. ∀[a:X(I)]. ∀[J:cat-ob(C)].
∀[f:cat-arrow(C) J I].
((u(a) a f) = u(f(a)) ∈ A(f(a)))
Proof
Definitions occuring in Statement :
presheaf-term-at: u(a),
presheaf-term: {X ⊢ _:A},
presheaf-type-ap-morph: (u a f),
presheaf-type-at: A(a),
presheaf-type: {X ⊢ _},
psc-restriction: f(s),
I_set: A(I),
ps_context: __⊢,
uall: ∀[x:A]. B[x],
apply: f a,
equal: s = t ∈ T,
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
presheaf-type: {X ⊢ _},
presheaf-term: {X ⊢ _:A},
all: ∀x:A. B[x],
presheaf-term-at: u(a),
subtype_rel: A ⊆r B
Lemmas referenced :
presheaf_type_at_pair_lemma,
presheaf_type_ap_morph_pair_lemma,
cat-arrow_wf,
I_set_wf,
cat-ob_wf,
presheaf-term_wf,
presheaf-type_wf,
ps_context_wf,
small-category-cumulativity-2,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
sqequalRule,
extract_by_obid,
dependent_functionElimination,
Error :memTop,
hypothesis,
hypothesisEquality,
universeIsType,
applyEquality,
isectElimination,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
instantiate
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[u:\{X \mvdash{} \_:A\}]. \mforall{}[I:cat-ob(C)].
\mforall{}[a:X(I)]. \mforall{}[J:cat-ob(C)]. \mforall{}[f:cat-arrow(C) J I].
((u(a) a f) = u(f(a)))
Date html generated:
2020_05_20-PM-01_26_41
Last ObjectModification:
2020_04_01-PM-01_51_05
Theory : presheaf!models!of!type!theory
Home
Index