Nuprl Lemma : presheaf-type-rev-iso_wf
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. (presheaf-type-rev-iso(X) ∈ {X ⊢ _} ⟶ presheaf_type{i:l}(C; X))
Proof
Definitions occuring in Statement :
presheaf-type-rev-iso: presheaf-type-rev-iso(X),
presheaf-type: {X ⊢ _},
presheaf_type: presheaf_type{i:l}(C; X),
ps_context: __⊢,
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
presheaf-type: {X ⊢ _},
presheaf-type-rev-iso: presheaf-type-rev-iso(X),
presheaf_type: presheaf_type{i:l}(C; X),
pi1: fst(t),
pi2: snd(t),
mk-presheaf: mk-presheaf,
presheaf: Presheaf(C),
subtype_rel: A ⊆r B,
and: P ∧ Q,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
cat-ob: cat-ob(C),
type-cat: TypeCat,
uimplies: b supposing a,
I_set: A(I),
ps_context: __⊢,
functor-ob: ob(F),
compose: f o g,
guard: {T},
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
prop: ℙ,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
presheaf-type_wf,
ps_context_wf,
small-category-cumulativity-2,
small-category_wf,
op-cat_wf,
sets_wf,
ps_context_cumulativity2,
type-cat_wf,
cat_ob_op_lemma,
sets-ob,
pi1_wf_top,
cat-ob_wf,
pi2_wf,
I_set_wf,
subtype_rel_universe1,
op-cat-arrow,
sets-arrow,
cat_arrow_triple_lemma,
subtype_rel-equal,
psc-restriction_wf,
cat-arrow_wf,
functor-ob_wf,
subtype_rel_self,
op-cat-comp,
sets-comp,
cat_comp_tuple_lemma,
op-cat-id,
sets-id,
cat_id_tuple_lemma,
mk-functor_wf,
psc-restriction-comp,
equal_wf,
cat-comp_wf,
squash_wf,
true_wf,
istype-universe,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
functionExtensionality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
sqequalRule,
hypothesis,
extract_by_obid,
isectElimination,
hypothesisEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
instantiate,
applyEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
dependent_functionElimination,
Error :memTop,
independent_pairEquality,
cumulativity,
lambdaEquality_alt,
because_Cache,
productIsType,
independent_isectElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
equalityIstype,
applyLambdaEquality,
setIsType,
universeEquality,
lambdaFormation_alt,
hyp_replacement,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)].
(presheaf-type-rev-iso(X) \mmember{} \{X \mvdash{} \_\} {}\mrightarrow{} presheaf\_type\{i:l\}(C; X))
Date html generated:
2020_05_20-PM-01_25_37
Last ObjectModification:
2020_04_02-AM-10_47_27
Theory : presheaf!models!of!type!theory
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