Nuprl Lemma : sub_ps_context_functionality
∀[C:SmallCategory]. ∀[Y,X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}].
sub_ps_context{[i | j]:l}(C; Y.A; X.A) supposing sub_ps_context{j:l}(C; Y; X)
Proof
Definitions occuring in Statement :
psc-adjoin: X.A,
presheaf-type: {X ⊢ _},
sub_ps_context: Y ⊆ X,
ps_context: __⊢,
uimplies: b supposing a,
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,
psc-adjoin: X.A,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
istype: istype(T),
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
sub_ps_context: Y ⊆ X
Lemmas referenced :
implies-sub_ps_context,
psc-adjoin_wf,
small-category-cumulativity-2,
ps_context_cumulativity2,
presheaf-type-cumulativity2,
I_set_pair_redex_lemma,
subtype_rel_product,
I_set_wf,
presheaf-type-at_wf,
subset-presheaf-type,
subset-I_set,
cat-ob_wf,
psc_restriction_pair_lemma,
equal_wf,
squash_wf,
true_wf,
istype-universe,
ps-subset-restriction,
pi1_wf_top,
psc-restriction_wf,
subtype_rel_self,
iff_weakening_equal,
presheaf-type-ap-morph_wf,
pi2_wf,
cat-arrow_wf,
sub_ps_context_wf,
presheaf-type_wf,
ps_context_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
Error :memTop,
lambdaFormation_alt,
cumulativity,
lambdaEquality_alt,
universeIsType,
dependent_pairEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
productElimination,
independent_pairEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
productIsType,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[Y,X:ps\_context\{j:l\}(C)]. \mforall{}[A:\{X \mvdash{} \_\}].
sub\_ps\_context\{[i | j]:l\}(C; Y.A; X.A) supposing sub\_ps\_context\{j:l\}(C; Y; X)
Date html generated:
2020_05_20-PM-01_35_27
Last ObjectModification:
2020_04_03-AM-01_23_09
Theory : presheaf!models!of!type!theory
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