Nuprl Lemma : presheaf-fun-eta
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A,B:{X ⊢ _}]. ∀[w:{X ⊢ _:(A ⟶ B)}].
(presheaf-lam(X;app((w)p; q)) = w ∈ {X ⊢ _:(A ⟶ B)})
Proof
Definitions occuring in Statement :
presheaf-app: app(w; u),
presheaf-lam: presheaf-lam(X;b),
presheaf-fun: (A ⟶ B),
psc-snd: q,
psc-fst: p,
pscm-ap-term: (t)s,
presheaf-term: {X ⊢ _:A},
presheaf-type: {X ⊢ _},
ps_context: __⊢,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
presheaf-lam: presheaf-lam(X;b),
member: t ∈ T,
subtype_rel: A ⊆r B,
squash: ↓T,
prop: ℙ,
true: True
Lemmas referenced :
presheaf-term_wf,
presheaf-fun_wf,
presheaf-type_wf,
ps_context_wf,
small-category-cumulativity-2,
small-category_wf,
squash_wf,
true_wf,
presheaf-fun-as-presheaf-pi,
presheaf-type-cumulativity2,
ps_context_cumulativity2,
presheaf-eta,
pscm-ap-type_wf,
psc-adjoin_wf,
psc-fst_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
instantiate,
applyEquality,
sqequalRule,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A,B:\{X \mvdash{} \_\}]. \mforall{}[w:\{X \mvdash{} \_:(A {}\mrightarrow{} B)\}].
(presheaf-lam(X;app((w)p; q)) = w)
Date html generated:
2020_05_20-PM-01_33_46
Last ObjectModification:
2020_04_03-AM-01_04_51
Theory : presheaf!models!of!type!theory
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