Nuprl Lemma : presheaf-app_wf_fun
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A,B:{X ⊢ _}]. ∀[w:{X ⊢ _:(A ⟶ B)}]. ∀[u:{X ⊢ _:A}].
(app(w; u) ∈ {X ⊢ _:B})
Proof
Definitions occuring in Statement :
presheaf-app: app(w; u),
presheaf-fun: (A ⟶ B),
presheaf-term: {X ⊢ _:A},
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,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
and: P ∧ Q,
squash: ↓T,
true: True
Lemmas referenced :
presheaf-fun-as-presheaf-pi,
presheaf-app_wf,
pscm-ap-type_wf,
ps_context_cumulativity2,
small-category-cumulativity-2,
psc-adjoin_wf,
presheaf-type-cumulativity2,
psc-fst_wf,
subtype_rel-equal,
presheaf-term_wf,
presheaf-fun_wf,
presheaf-pi_wf,
pscm-ap-type-fst-id-adjoin,
presheaf-type_wf,
ps_context_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
instantiate,
applyEquality,
hypothesis,
sqequalRule,
independent_isectElimination,
equalitySymmetry,
dependent_set_memberEquality_alt,
independent_pairFormation,
equalityTransitivity,
productIsType,
equalityIstype,
inhabitedIsType,
applyLambdaEquality,
setElimination,
rename,
productElimination,
lambdaEquality_alt,
imageElimination,
Error :memTop,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
universeIsType,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A,B:\{X \mvdash{} \_\}]. \mforall{}[w:\{X \mvdash{} \_:(A {}\mrightarrow{} B)\}]. \mforall{}[u:\{X \mvdash{} \_:A\}].
(app(w; u) \mmember{} \{X \mvdash{} \_:B\})
Date html generated:
2020_05_20-PM-01_31_16
Last ObjectModification:
2020_04_02-PM-05_54_03
Theory : presheaf!models!of!type!theory
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