Nuprl Lemma : ext-equal-presheaves-equiv-rel
∀[C:SmallCategory]. EquivRel(Presheaf(C);F,G.ext-equal-presheaves(C;F;G))
Proof
Definitions occuring in Statement :
ext-equal-presheaves: ext-equal-presheaves(C;F;G),
presheaf: Presheaf(C),
small-category: SmallCategory,
equiv_rel: EquivRel(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
equiv_rel: EquivRel(T;x,y.E[x; y]),
and: P ∧ Q,
refl: Refl(T;x,y.E[x; y]),
all: ∀x:A. B[x],
cand: A c∧ B,
sym: Sym(T;x,y.E[x; y]),
implies: P ⇒ Q,
prop: ℙ,
trans: Trans(T;x,y.E[x; y]),
ext-equal-presheaves: ext-equal-presheaves(C;F;G),
ext-eq: A ≡ B,
subtype_rel: A ⊆r B,
presheaf: Presheaf(C),
uimplies: b supposing a,
top: Top,
cat-arrow: cat-arrow(C),
pi1: fst(t),
pi2: snd(t),
type-cat: TypeCat,
cat-ob: cat-ob(C),
guard: {T}
Lemmas referenced :
presheaf_wf,
ext-equal-presheaves_wf,
cat-ob_wf,
cat-arrow_wf,
small-category_wf,
ext-eq_weakening,
functor-ob_wf,
op-cat_wf,
small-category-subtype,
type-cat_wf,
subtype_rel-equal,
cat_ob_op_lemma,
functor-arrow_wf,
op-cat-arrow,
subtype_rel_self,
subtype_rel_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
sqequalRule,
productElimination,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
instantiate,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
functionEquality,
functionExtensionality,
applyLambdaEquality,
universeEquality
Latex:
\mforall{}[C:SmallCategory]. EquivRel(Presheaf(C);F,G.ext-equal-presheaves(C;F;G))
Date html generated:
2020_05_20-AM-07_52_50
Last ObjectModification:
2017_10_03-PM-02_53_58
Theory : small!categories
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