Nuprl Lemma : presheaf-term-equal2
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[u,z:{X ⊢ _:A}].
u = z ∈ {X ⊢ _:A} supposing ∀I:cat-ob(C). ∀a:X(I). ((u I a) = (z I a) ∈ A(a))
Proof
Definitions occuring in Statement :
presheaf-term: {X ⊢ _:A},
presheaf-type-at: A(a),
presheaf-type: {X ⊢ _},
I_set: A(I),
ps_context: __⊢,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
equal: s = t ∈ T,
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
presheaf-term: {X ⊢ _:A},
all: ∀x:A. B[x],
presheaf-term-at: u(a),
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
I_set_wf,
cat-ob_wf,
presheaf-term-at-morph,
cat-arrow_wf,
presheaf-type-at_wf,
psc-restriction_wf,
presheaf-type-ap-morph_wf,
small-category-cumulativity-2,
ps_context_cumulativity2,
presheaf-term_wf,
presheaf-type_wf,
ps_context_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
dependent_set_memberEquality_alt,
functionExtensionality_alt,
universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
lambdaFormation_alt,
sqequalRule,
applyEquality,
because_Cache,
functionIsType,
equalityIstype,
instantiate,
setElimination,
rename,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
dependent_functionElimination
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[u,z:\{X \mvdash{} \_:A\}].
u = z supposing \mforall{}I:cat-ob(C). \mforall{}a:X(I). ((u I a) = (z I a))
Date html generated:
2020_05_20-PM-01_26_47
Last ObjectModification:
2020_04_01-PM-01_54_45
Theory : presheaf!models!of!type!theory
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