Nuprl Lemma : stable-element-predicate_wf
∀[C:SmallCategory]. ∀[F:Presheaf(C)]. ∀[P:I:cat-ob(C) ⟶ (ob(F) I) ⟶ ℙ].
(stable-element-predicate(C;F;I,rho.P[I;rho]) ∈ ℙ)
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]),
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
presheaf: Presheaf(C),
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s1;s2],
top: Top,
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 :
all_wf,
cat-ob_wf,
cat-arrow_wf,
functor-ob_wf,
op-cat_wf,
small-category-subtype,
type-cat_wf,
subtype_rel-equal,
cat_ob_op_lemma,
functor-arrow_wf,
op-cat-arrow,
subtype_rel_self,
presheaf_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesis,
lambdaEquality,
applyEquality,
hypothesisEquality,
instantiate,
independent_isectElimination,
dependent_functionElimination,
functionEquality,
functionExtensionality,
universeEquality,
isect_memberEquality,
voidElimination,
voidEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
cumulativity
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[F:Presheaf(C)]. \mforall{}[P:I:cat-ob(C) {}\mrightarrow{} (ob(F) I) {}\mrightarrow{} \mBbbP{}].
(stable-element-predicate(C;F;I,rho.P[I;rho]) \mmember{} \mBbbP{})
Date html generated:
2017_10_05-AM-00_50_54
Last ObjectModification:
2017_10_03-PM-03_11_33
Theory : small!categories
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