Nuprl Lemma : presheaf-subset-and
∀[C:SmallCategory]. ∀[F:Presheaf(C)]. ∀[P,Q:I:cat-ob(C) ⟶ (ob(F) I) ⟶ ℙ].
ext-equal-presheaves(C;F|I,rho.P[I;rho]|I,rho.Q[I;rho];F|I,rho.P[I;rho] ∧ Q[I;rho])
supposing stable-element-predicate(C;F;I,rho.P[I;rho]) ∧ stable-element-predicate(C;F;I,rho.Q[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]),
ext-equal-presheaves: ext-equal-presheaves(C;F;G),
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],
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
ext-equal-presheaves: ext-equal-presheaves(C;F;G),
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,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
presheaf: Presheaf(C),
cat-ob: cat-ob(C),
pi1: fst(t),
type-cat: TypeCat,
implies: P ⇒ Q,
cand: A c∧ B,
cat-arrow: cat-arrow(C),
pi2: snd(t),
guard: {T},
stable-element-predicate: stable-element-predicate(C;F;I,rho.P[I; rho])
Lemmas referenced :
ob_mk_functor_lemma,
arrow_mk_functor_lemma,
cat-arrow_wf,
cat-ob_wf,
stable-element-predicate_wf,
functor-ob_wf,
op-cat_wf,
small-category-subtype,
type-cat_wf,
subtype_rel-equal,
cat_ob_op_lemma,
presheaf_wf,
small-category_wf,
subtype_rel_sets,
subtype_rel_self,
functor-arrow_wf,
op-cat-arrow
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
lambdaFormation,
sqequalRule,
extract_by_obid,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
because_Cache,
applyEquality,
isectElimination,
hypothesisEquality,
independent_pairEquality,
lambdaEquality,
axiomEquality,
productEquality,
functionExtensionality,
instantiate,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality,
setElimination,
rename,
dependent_set_memberEquality,
setEquality,
independent_functionElimination
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[F:Presheaf(C)]. \mforall{}[P,Q:I:cat-ob(C) {}\mrightarrow{} (ob(F) I) {}\mrightarrow{} \mBbbP{}].
ext-equal-presheaves(C;F|I,rho.P[I;rho]|I,rho.Q[I;rho];F|I,rho.P[I;rho] \mwedge{} Q[I;rho])
supposing stable-element-predicate(C;F;I,rho.P[I;rho])
\mwedge{} stable-element-predicate(C;F;I,rho.Q[I;rho])
Date html generated:
2017_10_05-AM-00_51_10
Last ObjectModification:
2017_10_03-PM-03_30_44
Theory : small!categories
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