Nuprl Lemma : presheaf-subset-true
∀[C:SmallCategory]. ∀[F:Presheaf(C)]. ext-equal-presheaves(C;F|True;F)
Proof
Definitions occuring in Statement :
presheaf-subset: F|I,rho.P[I; rho],
ext-equal-presheaves: ext-equal-presheaves(C;F;G),
presheaf: Presheaf(C),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
true: True
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
ext-equal-presheaves: ext-equal-presheaves(C;F;G),
and: P ∧ Q,
all: ∀x:A. B[x],
presheaf-subset: F|I,rho.P[I; rho],
mk-presheaf: mk-presheaf,
top: Top,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
so_lambda: λ2x.t[x],
so_apply: x[s],
ext-eq: A ≡ B,
subtype_rel: A ⊆r B,
presheaf: Presheaf(C),
uimplies: b supposing a,
prop: ℙ,
true: True,
functor-arrow: arrow(F),
pi2: snd(t),
cat-arrow: cat-arrow(C),
pi1: fst(t),
type-cat: TypeCat
Lemmas referenced :
ob_mk_functor_lemma,
cat-ob_wf,
arrow_mk_functor_lemma,
cat-arrow_wf,
presheaf_wf,
small-category_wf,
functor-ob_wf,
op-cat_wf,
small-category-subtype,
type-cat_wf,
subtype_rel-equal,
cat_ob_op_lemma,
true_wf,
functor-arrow_wf,
op-cat-arrow,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
productElimination,
independent_pairEquality,
lambdaEquality,
axiomEquality,
setElimination,
rename,
setEquality,
instantiate,
independent_isectElimination,
dependent_set_memberEquality,
natural_numberEquality,
functionExtensionality,
functionEquality
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[F:Presheaf(C)]. ext-equal-presheaves(C;F|True;F)
Date html generated:
2017_10_05-AM-00_51_06
Last ObjectModification:
2017_10_03-PM-03_22_30
Theory : small!categories
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