Nuprl Lemma : presheaf-type-cumulativity2
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ({X ⊢ _} ⊆r presheaf-type{[j | i]:l}(C; X))
Proof
Definitions occuring in Statement :
presheaf-type: {X ⊢ _},
ps_context: __⊢,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
presheaf-type: {X ⊢ _},
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
all: ∀x:A. B[x],
and: P ∧ Q,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
subtype_rel_dep_function,
I_set_wf,
cat-ob_wf,
cat-arrow_wf,
psc-restriction_wf,
cat-id_wf,
subtype_rel-equal,
equal_wf,
squash_wf,
true_wf,
istype-universe,
psc-restriction-id,
subtype_rel_self,
iff_weakening_equal,
cat-comp_wf,
psc-restriction-comp,
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,
lambdaEquality_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality_alt,
productElimination,
dependent_pairEquality_alt,
functionExtensionality,
applyEquality,
hypothesisEquality,
hypothesis,
instantiate,
extract_by_obid,
isectElimination,
cumulativity,
sqequalRule,
universeEquality,
universeIsType,
because_Cache,
independent_isectElimination,
lambdaFormation_alt,
functionEquality,
productIsType,
functionIsType,
equalityIstype,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
dependent_functionElimination,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. (\{X \mvdash{} \_\} \msubseteq{}r presheaf-type\{[j | i]:l\}(C; X))
Date html generated:
2020_05_20-PM-01_25_49
Last ObjectModification:
2020_04_02-PM-00_41_34
Theory : presheaf!models!of!type!theory
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