Nuprl Lemma : subset-presheaf-type
∀[C:SmallCategory]. ∀[X,Y:ps_context{j:l}(C)]. {X ⊢ _} ⊆r {Y ⊢ _} supposing sub_ps_context{j:l}(C; Y; X)
Proof
Definitions occuring in Statement :
presheaf-type: {X ⊢ _},
sub_ps_context: Y ⊆ X,
ps_context: __⊢,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
sub_ps_context: Y ⊆ X,
presheaf-type: {X ⊢ _},
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
pscm-id: 1(X),
pscm-ap-type: (AF)s,
pscm-ap: (s)x
Lemmas referenced :
subset-I_set,
ps-subset-restriction,
presheaf-type-equal,
pscm-ap-type_wf,
subtype_rel_product,
cat-ob_wf,
I_set_wf,
cat-arrow_wf,
psc-restriction_wf,
istype-universe,
subtype_rel_dep_function,
subtype_rel-equal,
equal_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
presheaf-type_wf,
sub_ps_context_wf,
ps_context_wf,
small-category-cumulativity-2,
small-category_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
lambdaEquality_alt,
setElimination,
rename,
applyEquality,
instantiate,
functionEquality,
cumulativity,
universeEquality,
sqequalRule,
functionIsType,
universeIsType,
because_Cache,
lambdaFormation_alt,
dependent_functionElimination,
inhabitedIsType,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
dependent_pairEquality_alt,
axiomEquality,
functionExtensionality,
hyp_replacement
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X,Y:ps\_context\{j:l\}(C)].
\{X \mvdash{} \_\} \msubseteq{}r \{Y \mvdash{} \_\} supposing sub\_ps\_context\{j:l\}(C; Y; X)
Date html generated:
2020_05_20-PM-01_35_00
Last ObjectModification:
2020_04_02-PM-06_34_43
Theory : presheaf!models!of!type!theory
Home
Index