Nuprl Lemma : subset-presheaf-term2
∀[C:SmallCategory]. ∀[X,Y:ps_context{j:l}(C)].
∀[A,B:{X ⊢ _}]. {X ⊢ _:A} ⊆r {Y ⊢ _:B} supposing A = B ∈ {X ⊢ _} supposing sub_ps_context{j:l}(C; Y; X)
Proof
Definitions occuring in Statement :
presheaf-term: {X ⊢ _:A},
presheaf-type: {X ⊢ _},
sub_ps_context: Y ⊆ X,
ps_context: __⊢,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
subset-presheaf-term,
presheaf-type_wf,
sub_ps_context_wf,
ps_context_wf,
small-category-cumulativity-2,
small-category_wf,
presheaf-term_wf,
subset-presheaf-type,
subtype_rel_wf,
squash_wf,
true_wf,
istype-universe,
presheaf-type-cumulativity2,
ps_context_cumulativity2,
iff_weakening_equal
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
sqequalRule,
axiomEquality,
equalityIstype,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
because_Cache,
universeIsType,
instantiate,
applyEquality,
natural_numberEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X,Y:ps\_context\{j:l\}(C)].
\mforall{}[A,B:\{X \mvdash{} \_\}]. \{X \mvdash{} \_:A\} \msubseteq{}r \{Y \mvdash{} \_:B\} supposing A = B supposing sub\_ps\_context\{j:l\}(C; Y; X)
Date html generated:
2020_05_20-PM-01_35_09
Last ObjectModification:
2020_04_03-AM-01_20_27
Theory : presheaf!models!of!type!theory
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