Nuprl Lemma : presheaf-subset_wf
∀[C:SmallCategory]. ∀[F:Presheaf(C)]. ∀[P:I:cat-ob(C) ⟶ (F I) ⟶ ℙ].
F|I,rho.P[I;rho] ∈ Presheaf(C) supposing stable-element-predicate(C;F;I,rho.P[I;rho])
Proof
Definitions occuring in Statement :
presheaf-subset: F|I,rho.P[I; rho],
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,
uimplies: b supposing a,
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,
uimplies: b supposing a,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
presheaf: Presheaf(C),
all: ∀x:A. B[x],
cat-ob: cat-ob(C),
pi1: fst(t),
type-cat: TypeCat,
prop: ℙ
Lemmas referenced :
presheaf-subset_wf1,
functor-ob_wf,
op-cat_wf,
small-category-subtype,
type-cat_wf,
subtype_rel-equal,
cat-ob_wf,
cat_ob_op_lemma,
subtype_rel_self,
stable-element-predicate_wf,
presheaf_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
applyEquality,
universeIsType,
hypothesis,
independent_isectElimination,
dependent_functionElimination,
universeEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
functionIsType
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[F:Presheaf(C)]. \mforall{}[P:I:cat-ob(C) {}\mrightarrow{} (F I) {}\mrightarrow{} \mBbbP{}].
F|I,rho.P[I;rho] \mmember{} Presheaf(C) supposing stable-element-predicate(C;F;I,rho.P[I;rho])
Date html generated:
2020_05_20-AM-07_57_32
Last ObjectModification:
2020_04_03-AM-11_40_33
Theory : small!categories
Home
Index