Nuprl Lemma : presheaf-term-equal
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[u:{X ⊢ _:A}]. ∀[z:I:cat-ob(C) ⟶ a:X(I) ⟶ A(a)].
u = z ∈ {X ⊢ _:A} supposing u = z ∈ (I:cat-ob(C) ⟶ a:X(I) ⟶ 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],
function: x:A ⟶ B[x],
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
Lemmas referenced :
presheaf-term-at-morph,
I_set_wf,
cat-arrow_wf,
cat-ob_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,
hypothesis,
lambdaFormation_alt,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
universeIsType,
applyEquality,
inhabitedIsType,
functionIsType,
because_Cache,
equalityIstype,
instantiate,
setElimination,
rename,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[u:\{X \mvdash{} \_:A\}]. \mforall{}[z:I:cat-ob(C)
{}\mrightarrow{} a:X(I)
{}\mrightarrow{} A(a)].
u = z supposing u = z
Date html generated:
2020_05_20-PM-01_26_44
Last ObjectModification:
2020_04_01-PM-01_54_42
Theory : presheaf!models!of!type!theory
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