Nuprl Lemma : ps_contexts_equal
∀[C:SmallCategory]. ∀[X,Y:ps_context{j:l}(C)].
X = Y ∈ ps_context{j:l}(C)
supposing X = Y ∈ (F:cat-ob(C) ⟶ 𝕌{j'} × (x:cat-ob(C) ⟶ y:cat-ob(C) ⟶ (cat-arrow(C) y x) ⟶ (F x) ⟶ (F y)))
Proof
Definitions occuring in Statement :
ps_context: __⊢,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T,
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
subtype_rel: A ⊆r B,
ps_context: __⊢,
cat-functor: Functor(C1;C2),
small-category: SmallCategory,
spreadn: spread4,
all: ∀x:A. B[x],
type-cat: TypeCat,
op-cat: op-cat(C),
so_lambda: λ2x.t[x],
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
and: P ∧ Q,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
equal-functors,
op-cat_wf,
type-cat_wf,
small-category-cumulativity-2,
cat_ob_pair_lemma,
cat_arrow_triple_lemma,
ob_pair_lemma,
cat_id_tuple_lemma,
cat_comp_tuple_lemma,
pi2_wf,
pi1_wf_top,
arrow_pair_lemma,
equal_wf,
squash_wf,
true_wf,
istype-universe,
subtype_rel_dep_function,
subtype_rel-equal,
subtype_rel_self,
iff_weakening_equal,
ps_context_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
sqequalRule,
independent_isectElimination,
setElimination,
rename,
productElimination,
dependent_functionElimination,
Error :memTop,
applyLambdaEquality,
functionEquality,
cumulativity,
universeEquality,
lambdaEquality_alt,
inhabitedIsType,
independent_pairEquality,
lambdaFormation_alt,
equalitySymmetry,
dependent_set_memberEquality_alt,
independent_pairFormation,
equalityTransitivity,
productIsType,
equalityIstype,
universeIsType,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
functionIsType,
dependent_pairEquality_alt
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X,Y:ps\_context\{j:l\}(C)]. X = Y supposing X = Y
Date html generated:
2020_05_20-PM-01_23_08
Last ObjectModification:
2020_04_02-PM-01_41_36
Theory : presheaf!models!of!type!theory
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