Nuprl Lemma : presheaf-type-equal
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[B:A:I:cat-ob(C) ⟶ X(I) ⟶ Type × (I:cat-ob(C)
⟶ J:cat-ob(C)
⟶ f:(cat-arrow(C) J I)
⟶ a:X(I)
⟶ (A I a)
⟶ (A J f(a)))].
A = B ∈ {X ⊢ _}
supposing A
= B
∈ (A:I:cat-ob(C) ⟶ X(I) ⟶ Type × (I:cat-ob(C)
⟶ J:cat-ob(C)
⟶ f:(cat-arrow(C) J I)
⟶ a:X(I)
⟶ (A I a)
⟶ (A J f(a))))
Proof
Definitions occuring in Statement :
presheaf-type: {X ⊢ _},
psc-restriction: f(s),
I_set: A(I),
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],
member: t ∈ T,
uimplies: b supposing a,
presheaf-type: {X ⊢ _},
and: P ∧ Q,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
squash: ↓T,
true: True,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
cat-ob_wf,
I_set_wf,
cat-id_wf,
subtype_rel-equal,
psc-restriction_wf,
equal_wf,
psc-restriction-id,
ps_context_cumulativity2,
subtype_rel_self,
iff_weakening_equal,
cat-arrow_wf,
cat-comp_wf,
psc-restriction-comp,
small-category-cumulativity-2,
istype-universe,
presheaf-type_wf,
ps_context_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality_alt,
hypothesis,
productElimination,
sqequalRule,
productIsType,
functionIsType,
universeIsType,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
equalityIstype,
because_Cache,
instantiate,
independent_isectElimination,
lambdaEquality_alt,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
dependent_functionElimination,
inhabitedIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A:\{X \mvdash{} \_\}].
\mforall{}[B:A:I:cat-ob(C) {}\mrightarrow{} X(I) {}\mrightarrow{} Type \mtimes{} (I:cat-ob(C)
{}\mrightarrow{} J:cat-ob(C)
{}\mrightarrow{} f:(cat-arrow(C) J I)
{}\mrightarrow{} a:X(I)
{}\mrightarrow{} (A I a)
{}\mrightarrow{} (A J f(a)))].
A = B supposing A = B
Date html generated:
2020_05_20-PM-01_25_20
Last ObjectModification:
2020_04_01-AM-11_00_46
Theory : presheaf!models!of!type!theory
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