Nuprl Lemma : presheaf-type_wf
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. (X ⊢ ∈ 𝕌{[j' | i']})
Proof
Definitions occuring in Statement :
presheaf-type: {X ⊢ _},
ps_context: __⊢,
uall: ∀[x:A]. B[x],
member: t ∈ T,
universe: Type,
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
presheaf-type: {X ⊢ _},
subtype_rel: A ⊆r B,
and: P ∧ Q,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
so_apply: x[s],
prop: ℙ,
all: ∀x:A. B[x]
Lemmas referenced :
cat-ob_wf,
I_set_wf,
cat-arrow_wf,
psc-restriction_wf,
all_wf,
equal_wf,
cat-id_wf,
subtype_rel-equal,
psc-restriction-id,
ps_context_cumulativity2,
iff_weakening_equal,
small-category-cumulativity-2,
cat-comp_wf,
psc-restriction-comp,
ps_context_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
setEquality,
productEquality,
functionEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality_alt,
cumulativity,
universeIsType,
universeEquality,
because_Cache,
closedConclusion,
instantiate,
productElimination,
independent_isectElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
inhabitedIsType,
dependent_functionElimination,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. (X \mvdash{} \mmember{} \mBbbU{}\{[j' | i']\})
Date html generated:
2020_05_20-PM-01_25_17
Last ObjectModification:
2020_03_31-PM-02_37_08
Theory : presheaf!models!of!type!theory
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