Nuprl Lemma : presheaf-app-id-fun
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[u:{X ⊢ _:A}]. (app(presheaf-id-fun(X); u) = u ∈ {X ⊢ _:A})
Proof
Definitions occuring in Statement :
presheaf-app: app(w; u),
presheaf-id-fun: presheaf-id-fun(X),
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],
member: t ∈ T,
presheaf-id-fun: presheaf-id-fun(X),
presheaf-lam: presheaf-lam(X;b),
subtype_rel: A ⊆r B,
prop: ℙ,
squash: ↓T,
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
presheaf-beta,
pscm-ap-type_wf,
ps_context_cumulativity2,
small-category-cumulativity-2,
psc-adjoin_wf,
presheaf-type-cumulativity2,
psc-fst_wf,
psc-snd_wf,
presheaf-term_wf,
presheaf-type_wf,
ps_context_wf,
small-category_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
pscm-ap-type-fst-id-adjoin,
iff_weakening_equal,
subtype_rel-equal,
pscm-id-adjoin_wf,
subtype_rel_self,
ps-cc_snd_csm_id_adjoin_lemma,
pscm-ap-id-term
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
instantiate,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
universeIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
hyp_replacement,
equalitySymmetry,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
universeEquality,
Error :memTop,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
dependent_functionElimination
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[u:\{X \mvdash{} \_:A\}].
(app(presheaf-id-fun(X); u) = u)
Date html generated:
2020_05_20-PM-01_33_56
Last ObjectModification:
2020_04_03-AM-01_06_33
Theory : presheaf!models!of!type!theory
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