Nuprl Lemma : stable-element-predicate_wf1
∀[C:SmallCategory]. ∀[F:presheaf{j:l}(C)]. ∀[P:I:cat-ob(C) ⟶ (F I) ⟶ ℙ{j}].
(stable-element-predicate(C;F;I,rho.P[I;rho]) ∈ ℙ{[i | j]})
Proof
Definitions occuring in Statement :
stable-element-predicate: stable-element-predicate(C;F;I,rho.P[I; rho]),
presheaf: Presheaf(C),
functor-ob: ob(F),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
stable-element-predicate: stable-element-predicate(C;F;I,rho.P[I; rho]),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
presheaf: Presheaf(C),
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s1;s2],
cat-arrow: cat-arrow(C),
pi1: fst(t),
pi2: snd(t),
type-cat: TypeCat,
so_apply: x[s],
cat-ob: cat-ob(C)
Lemmas referenced :
presheaf_wf1,
all_wf,
cat-ob_wf,
cat-arrow_wf,
functor-ob_wf,
op-cat_wf,
small-category-cumulativity-2,
type-cat_wf,
subtype_rel-equal,
cat_ob_op_lemma,
functor-arrow_wf,
op-cat-arrow,
subtype_rel_self,
small-category_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
instantiate,
applyEquality,
lambdaEquality_alt,
cumulativity,
universeIsType,
universeEquality,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
functionEquality,
Error :memTop,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
functionIsType,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[F:presheaf\{j:l\}(C)]. \mforall{}[P:I:cat-ob(C) {}\mrightarrow{} (F I) {}\mrightarrow{} \mBbbP{}\{j\}].
(stable-element-predicate(C;F;I,rho.P[I;rho]) \mmember{} \mBbbP{}\{[i | j]\})
Date html generated:
2020_05_20-AM-07_57_19
Last ObjectModification:
2020_04_03-AM-11_18_30
Theory : small!categories
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