Nuprl Lemma : presheaf-subset-and
∀[C:SmallCategory]. ∀[F:presheaf{j:l}(C)]. ∀[P,Q:I:cat-ob(C) ⟶ (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,
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
so_lambda: λ2x.t[x],
so_apply: x[s],
ext-eq: A ≡ B,
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
prop: ℙ,
so_apply: x[s1;s2],
presheaf: Presheaf(C),
cat-ob: cat-ob(C),
pi1: fst(t),
type-cat: TypeCat,
guard: {T},
stable-element-predicate: stable-element-predicate(C;F;I,rho.P[I; rho]),
implies: P ⇒ Q
Lemmas referenced :
small-category_wf,
ob_mk_functor_lemma,
arrow_mk_functor_lemma,
cat-arrow_wf,
stable-element-predicate_wf,
small-category-cumulativity-2,
presheaf-cumulativity1,
functor-ob_wf,
op-cat_wf,
type-cat_wf,
subtype_rel-equal,
cat-ob_wf,
cat_ob_op_lemma,
subtype_rel_self,
cat_ob_pair_lemma,
functor-arrow_wf,
op-cat-arrow,
presheaf_wf1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
lambdaFormation_alt,
sqequalRule,
dependent_functionElimination,
Error :memTop,
because_Cache,
applyEquality,
isectElimination,
hypothesisEquality,
independent_pairEquality,
lambdaEquality_alt,
axiomEquality,
functionIsTypeImplies,
inhabitedIsType,
productIsType,
instantiate,
cumulativity,
independent_isectElimination,
universeEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
functionIsType,
setElimination,
rename,
dependent_set_memberEquality_alt,
setIsType,
functionExtensionality,
setEquality,
independent_functionElimination
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[F:presheaf\{j:l\}(C)]. \mforall{}[P,Q:I:cat-ob(C) {}\mrightarrow{} (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:
2020_05_20-AM-07_57_39
Last ObjectModification:
2020_04_02-AM-09_48_25
Theory : small!categories
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