Nuprl Lemma : subset-presheaf-term
∀[C:SmallCategory]. ∀[X,Y:ps_context{j:l}(C)].
∀[A:{X ⊢ _}]. ({X ⊢ _:A} ⊆r {Y ⊢ _:A}) 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],
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,
guard: {T},
squash: ↓T,
true: True,
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
pscm-id: 1(X),
pscm-ap-term: (t)s,
pscm-ap: (s)x,
presheaf-term-at: u(a),
all: ∀x:A. B[x]
Lemmas referenced :
subset-presheaf-type,
pscm-ap-term_wf,
presheaf-term_wf,
presheaf-type_wf,
sub_ps_context_wf,
ps_context_wf,
small-category-cumulativity-2,
small-category_wf,
I_set_wf,
cat-ob_wf,
presheaf-term-equal,
subtype_rel-equal,
pscm-ap-type_wf,
equal_wf,
pscm-ap-id-type,
subtype_rel_transitivity,
subtype_rel_self,
iff_weakening_equal,
subset-I_set,
presheaf-term-at_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,
equalityTransitivity,
equalitySymmetry,
universeIsType,
sqequalRule,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
applyEquality,
because_Cache,
functionExtensionality,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
dependent_functionElimination,
lambdaFormation_alt,
equalityIstype
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X,Y:ps\_context\{j:l\}(C)].
\mforall{}[A:\{X \mvdash{} \_\}]. (\{X \mvdash{} \_:A\} \msubseteq{}r \{Y \mvdash{} \_:A\}) supposing sub\_ps\_context\{j:l\}(C; Y; X)
Date html generated:
2020_05_20-PM-01_35_05
Last ObjectModification:
2020_04_03-AM-01_16_04
Theory : presheaf!models!of!type!theory
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