Nuprl Lemma : presheaf-fun-family_wf
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A,B:{X ⊢ _}]. ∀[I:cat-ob(C)]. ∀[a:X(I)].
(presheaf-fun-family(C; X; A; B; I; a) ∈ Type)
Proof
Definitions occuring in Statement :
presheaf-fun-family: presheaf-fun-family(C; X; A; B; I; a),
presheaf-type: {X ⊢ _},
I_set: A(I),
ps_context: __⊢,
uall: ∀[x:A]. B[x],
member: t ∈ T,
universe: Type,
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
presheaf-fun-family: presheaf-fun-family(C; X; A; B; I; a),
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
squash: ↓T,
true: True,
prop: ℙ
Lemmas referenced :
cat-ob_wf,
cat-arrow_wf,
presheaf-type-at_wf,
psc-restriction_wf,
equal_wf,
presheaf-type-ap-morph_wf,
cat-comp_wf,
subtype_rel-equal,
psc-restriction-comp,
I_set_wf,
presheaf-type_wf,
ps_context_wf,
small-category-cumulativity-2,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
setEquality,
functionEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
independent_isectElimination,
lambdaEquality_alt,
imageElimination,
because_Cache,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalitySymmetry,
axiomEquality,
equalityTransitivity,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
instantiate
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A,B:\{X \mvdash{} \_\}]. \mforall{}[I:cat-ob(C)]. \mforall{}[a:X(I)].
(presheaf-fun-family(C; X; A; B; I; a) \mmember{} Type)
Date html generated:
2020_05_20-PM-01_28_58
Last ObjectModification:
2020_04_02-PM-01_56_35
Theory : presheaf!models!of!type!theory
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