Nuprl Lemma : presheaf-apply_wf
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[w:{X ⊢ _:ΠA B}]. ∀[u:{X ⊢ _:A}].
(presheaf-apply(w;u) ∈ {X ⊢ _:(B)[u]})
Proof
Definitions occuring in Statement :
presheaf-apply: presheaf-apply(w;u),
presheaf-pi: ΠA B,
pscm-id-adjoin: [u],
psc-adjoin: X.A,
presheaf-term: {X ⊢ _:A},
pscm-ap-type: (AF)s,
presheaf-type: {X ⊢ _},
ps_context: __⊢,
uall: ∀[x:A]. B[x],
member: t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
presheaf-apply: presheaf-apply(w;u),
subtype_rel: A ⊆r B
Lemmas referenced :
presheaf-app_wf,
ps_context_cumulativity2,
presheaf-type-cumulativity2,
psc-adjoin_wf,
presheaf-term_wf,
presheaf-pi_wf,
presheaf-type_wf,
small-category-cumulativity-2,
ps_context_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[w:\{X \mvdash{} \_:\mPi{}A B\}].
\mforall{}[u:\{X \mvdash{} \_:A\}].
(presheaf-apply(w;u) \mmember{} \{X \mvdash{} \_:(B)[u]\})
Date html generated:
2020_05_20-PM-01_31_13
Last ObjectModification:
2020_04_02-PM-03_02_38
Theory : presheaf!models!of!type!theory
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