Nuprl Lemma : presheaf-subset_wf
∀[C:SmallCategory]. ∀[F:Presheaf(C)]. ∀[P:I:cat-ob(C) ⟶ (ob(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,
presheaf-subset: F|I,rho.P[I; rho],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
presheaf: Presheaf(C),
all: ∀x:A. B[x],
so_apply: x[s1;s2],
prop: ℙ,
so_apply: x[s],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
cat-ob: cat-ob(C),
pi1: fst(t),
type-cat: TypeCat,
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x y.t[x; y],
guard: {T},
stable-element-predicate: stable-element-predicate(C;F;I,rho.P[I; rho]),
implies: P ⇒ Q,
top: Top,
functor-arrow: arrow(F),
pi2: snd(t),
compose: f o g
Lemmas referenced :
mk-presheaf_wf,
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,
cat-arrow_wf,
set_wf,
stable-element-predicate_wf,
presheaf_wf,
small-category_wf,
functor-arrow-id,
op-cat-id,
cat_arrow_triple_lemma,
cat_id_tuple_lemma,
cat-comp_wf,
functor-arrow-comp,
op-cat-arrow,
op-cat-comp,
cat_comp_tuple_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
lambdaEquality,
setEquality,
applyEquality,
instantiate,
hypothesisEquality,
hypothesis,
independent_isectElimination,
dependent_functionElimination,
functionExtensionality,
universeEquality,
lambdaFormation,
setElimination,
rename,
dependent_set_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
functionEquality,
cumulativity,
independent_functionElimination,
voidElimination,
voidEquality,
applyLambdaEquality
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[F:Presheaf(C)]. \mforall{}[P:I:cat-ob(C) {}\mrightarrow{} (ob(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:
2017_10_05-AM-00_51_02
Last ObjectModification:
2017_10_03-PM-03_19_41
Theory : small!categories
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