Nuprl Lemma : equal-presheaves
∀[C:SmallCategory]. ∀[F,G:Presheaf(C)].
F = G ∈ Presheaf(C)
supposing (∀x:cat-ob(op-cat(C)). ((ob(F) x) = (ob(G) x) ∈ cat-ob(TypeCat)))
∧ (∀x,y:cat-ob(op-cat(C)). ∀f:cat-arrow(op-cat(C)) x y.
((arrow(F) x y f) = (arrow(G) x y f) ∈ (cat-arrow(TypeCat) (ob(F) x) (ob(F) y))))
Proof
Definitions occuring in Statement :
presheaf: Presheaf(C),
type-cat: TypeCat,
op-cat: op-cat(C),
functor-arrow: arrow(F),
functor-ob: ob(F),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
presheaf: Presheaf(C),
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
equal-functors,
op-cat_wf,
type-cat_wf,
all_wf,
cat-ob_wf,
equal_wf,
functor-ob_wf,
small-category-subtype,
cat-arrow_wf,
functor-arrow_wf,
subtype_rel-equal,
squash_wf,
true_wf,
iff_weakening_equal,
presheaf_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
instantiate,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
independent_isectElimination,
productEquality,
lambdaEquality,
cumulativity,
universeEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[F,G:Presheaf(C)].
F = G
supposing (\mforall{}x:cat-ob(op-cat(C)). ((ob(F) x) = (ob(G) x)))
\mwedge{} (\mforall{}x,y:cat-ob(op-cat(C)). \mforall{}f:cat-arrow(op-cat(C)) x y. ((arrow(F) x y f) = (arrow(G) x y f)))
Date html generated:
2017_10_05-AM-00_47_28
Last ObjectModification:
2017_10_03-PM-02_24_56
Theory : small!categories
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