Nuprl Lemma : member-presheaf-fun-p
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A,B,T:{X ⊢ _}]. ∀[x:{X ⊢ _:(A ⟶ B)}]. ((x)p ∈ {X.T ⊢ _:((A)p ⟶ (B)p)})
Proof
Definitions occuring in Statement :
presheaf-fun: (A ⟶ B),
psc-fst: p,
psc-adjoin: X.A,
pscm-ap-term: (t)s,
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,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
squash: ↓T
Lemmas referenced :
pscm-ap-term_wf,
psc-adjoin_wf,
ps_context_cumulativity2,
presheaf-type-cumulativity2,
presheaf-fun_wf,
psc-fst_wf,
subtype_rel-equal,
presheaf-term_wf,
pscm-ap-type_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
because_Cache,
applyEquality,
hypothesis,
sqequalRule,
independent_isectElimination,
lambdaEquality_alt,
imageElimination,
imageMemberEquality,
baseClosed,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A,B,T:\{X \mvdash{} \_\}]. \mforall{}[x:\{X \mvdash{} \_:(A {}\mrightarrow{} B)\}].
((x)p \mmember{} \{X.T \mvdash{} \_:((A)p {}\mrightarrow{} (B)p)\})
Date html generated:
2020_05_20-PM-01_29_57
Last ObjectModification:
2020_04_02-PM-05_58_44
Theory : presheaf!models!of!type!theory
Home
Index