Nuprl Lemma : ps-canonical-section_wf
∀[C:SmallCategory]. ∀[Gamma:ps_context{j:l}(C)]. ∀[A:{Gamma ⊢ _}]. ∀[I:cat-ob(C)]. ∀[rho:Gamma(I)]. ∀[a:A(rho)].
(ps-canonical-section(Gamma;A;I;rho;a) ∈ {Yoneda(I) ⊢ _:(A)<rho>})
Proof
Definitions occuring in Statement :
ps-canonical-section: ps-canonical-section(Gamma;A;I;rho;a),
presheaf-term: {X ⊢ _:A},
pscm-ap-type: (AF)s,
presheaf-type-at: A(a),
presheaf-type: {X ⊢ _},
ps-context-map: <rho>,
Yoneda: Yoneda(I),
I_set: A(I),
ps_context: __⊢,
uall: ∀[x:A]. B[x],
member: t ∈ T,
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
presheaf-term: {X ⊢ _:A},
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
Yoneda: Yoneda(I),
ps-canonical-section: ps-canonical-section(Gamma;A;I;rho;a),
cat-arrow: cat-arrow(C),
pi1: fst(t),
pi2: snd(t),
I_set: A(I),
functor-ob: ob(F),
uimplies: b supposing a,
squash: ↓T,
psc-restriction: f(s),
pscm-ap: (s)x,
ps-context-map: <rho>,
functor-arrow: arrow(F),
true: True
Lemmas referenced :
pscm-ap-type-at,
I_set_wf,
Yoneda_wf,
cat-arrow_wf,
presheaf-type-at_wf,
pscm-ap-type_wf,
ps-context-map_wf,
psc-restriction_wf,
presheaf-type-ap-morph_wf,
ps_context_cumulativity2,
small-category-cumulativity-2,
cat-ob_wf,
presheaf-type_wf,
ps_context_wf,
small-category_wf,
I_set_pair_redex_lemma,
subtype_rel-equal,
subtype_rel_self,
pscm-ap_wf,
psc_restriction_pair_lemma,
pscm-presheaf-type-ap-morph,
pscm-ap-context-map,
presheaf-type-ap-morph-comp
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
dependent_set_memberEquality_alt,
lambdaFormation_alt,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
Error :memTop,
hypothesis,
universeIsType,
instantiate,
hypothesisEquality,
applyEquality,
because_Cache,
functionIsType,
equalityIstype,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
lambdaEquality_alt,
independent_isectElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
rename
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[Gamma:ps\_context\{j:l\}(C)]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[I:cat-ob(C)]. \mforall{}[rho:Gamma(I)].
\mforall{}[a:A(rho)].
(ps-canonical-section(Gamma;A;I;rho;a) \mmember{} \{Yoneda(I) \mvdash{} \_:(A)<rho>\})
Date html generated:
2020_05_20-PM-01_26_51
Last ObjectModification:
2020_04_01-PM-00_01_21
Theory : presheaf!models!of!type!theory
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