Nuprl Lemma : presheaf-type-equal3
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A,B:{X ⊢ _}].
(A = B ∈ {X ⊢ _}) supposing
((∀I:cat-ob(C)
∀[rho:X(I)]. ∀[J:cat-ob(C)]. ∀[f:cat-arrow(C) J I]. ∀[u:A(rho)]. ((u rho f) = (u rho f) ∈ A(f(rho)))) and
(∀I:cat-ob(C). ∀[rho:X(I)]. (A(rho) = B(rho) ∈ Type)))
Proof
Definitions occuring in Statement :
presheaf-type-ap-morph: (u a f),
presheaf-type-at: A(a),
presheaf-type: {X ⊢ _},
psc-restriction: f(s),
I_set: A(I),
ps_context: __⊢,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
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],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
presheaf-type: {X ⊢ _},
and: P ∧ Q,
guard: {T},
prop: ℙ
Lemmas referenced :
presheaf-type-equal2,
cat-ob_wf,
I_set_wf,
istype-universe,
presheaf-type-at_wf,
presheaf-type_wf,
ps_context_wf,
small-category-cumulativity-2,
small-category_wf,
presheaf_type_at_pair_lemma,
presheaf_type_ap_morph_pair_lemma,
cat-arrow_wf,
psc-restriction_wf,
ps_context_cumulativity2,
equal_wf,
presheaf-type-ap-morph_wf,
subtype_rel-equal
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
functionIsType,
universeIsType,
isectIsType,
equalityIstype,
instantiate,
universeEquality,
inhabitedIsType,
applyEquality,
independent_isectElimination,
setElimination,
rename,
productElimination,
dependent_pairEquality_alt,
functionExtensionality,
dependent_functionElimination,
Error :memTop,
because_Cache,
equalityTransitivity,
equalitySymmetry,
lambdaEquality_alt,
hyp_replacement
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A,B:\{X \mvdash{} \_\}].
(A = B) supposing
((\mforall{}I:cat-ob(C)
\mforall{}[rho:X(I)]. \mforall{}[J:cat-ob(C)]. \mforall{}[f:cat-arrow(C) J I]. \mforall{}[u:A(rho)].
((u rho f) = (u rho f))) and
(\mforall{}I:cat-ob(C). \mforall{}[rho:X(I)]. (A(rho) = B(rho))))
Date html generated:
2020_05_20-PM-01_26_09
Last ObjectModification:
2020_04_01-PM-00_01_26
Theory : presheaf!models!of!type!theory
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