Nuprl Lemma : presheaf-app_wf
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[w:{X ⊢ _:ΠA B}]. ∀[u:{X ⊢ _:A}].
(app(w; u) ∈ {X ⊢ _:(B)[u]})
Proof
Definitions occuring in Statement :
presheaf-app: app(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-term: {X ⊢ _:A},
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
presheaf-type: {X ⊢ _},
presheaf-app: app(w; u),
pscm-ap-type: (AF)s,
presheaf-type-at: A(a),
pi1: fst(t),
I_set: A(I),
functor-ob: ob(F),
psc-adjoin: X.A,
psc-adjoin-set: (v;u),
pscm-ap: (s)x,
pscm-id-adjoin: [u],
pscm-adjoin: (s;u),
squash: ↓T,
prop: ℙ,
uimplies: b supposing a,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
pscm-id: 1(X),
presheaf-pi: ΠA B,
presheaf-pi-family: presheaf-pi-family(C; X; A; B; I; a),
presheaf-type-ap-morph: (u a f),
pi2: snd(t),
psc-restriction: f(s),
so_lambda: λ2x.t[x],
so_apply: x[s],
istype: istype(T)
Lemmas referenced :
pscm-ap-type_wf,
psc-adjoin_wf,
ps_context_cumulativity2,
presheaf-type-cumulativity2,
pscm-id-adjoin_wf,
presheaf_type_at_pair_lemma,
presheaf_type_ap_morph_pair_lemma,
cat-ob_wf,
cat-arrow_wf,
I_set_wf,
psc-restriction_wf,
presheaf-term_wf,
small-category-cumulativity-2,
presheaf-pi_wf,
presheaf-type_wf,
ps_context_wf,
small-category_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
psc-restriction-when-id,
cat-id_wf,
pscm-ap_wf,
pscm-id_wf,
subtype_rel_self,
iff_weakening_equal,
presheaf-type-at_wf,
subtype_rel-equal,
psc-restriction-id,
subtype_rel_weakening,
ext-eq_weakening,
psc-adjoin-set_wf,
subtype_rel_wf,
psc-adjoin-set-restriction,
cat-comp-ident,
subtype_rel_dep_function,
cat-comp_wf,
cat-comp-ident1,
cat-comp-ident2,
subtype_rel_set,
equal_functionality_wrt_subtype_rel2,
pscm-id-adjoin-ap,
pscm-ap-restriction
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
dependent_set_memberEquality_alt,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
inhabitedIsType,
lambdaFormation_alt,
setElimination,
rename,
productElimination,
dependent_functionElimination,
Error :memTop,
functionIsType,
universeIsType,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
lambdaEquality_alt,
dependent_pairEquality_alt,
imageElimination,
universeEquality,
independent_isectElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
promote_hyp,
applyLambdaEquality,
functionEquality,
hyp_replacement,
independent_pairFormation,
productIsType,
productEquality,
setEquality
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\}].
(app(w; u) \mmember{} \{X \mvdash{} \_:(B)[u]\})
Date html generated:
2020_05_20-PM-01_31_05
Last ObjectModification:
2020_04_02-PM-03_05_49
Theory : presheaf!models!of!type!theory
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