Nuprl Lemma : psc-adjoin-wf
∀[C:SmallCategory]. ∀[Gamma:ps_context{j:l}(C)]. ∀[A:{Gamma ⊢' _}]. (Gamma.A ∈ ps_context{[i | j]:l}(C))
Proof
Definitions occuring in Statement :
psc-adjoin: X.A,
presheaf-type: {X ⊢ _},
ps_context: __⊢,
uall: ∀[x:A]. B[x],
member: t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
ext-eq: A ≡ B,
and: P ∧ Q,
psc-adjoin: X.A,
subtype_rel: A ⊆r B,
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
squash: ↓T,
prop: ℙ,
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
psc-restriction: f(s)
Lemmas referenced :
ps_context-ext,
I_set_wf,
presheaf-type-at_wf,
cat-ob_wf,
psc-restriction_wf,
presheaf-type-ap-morph_wf,
cat-arrow_wf,
I_set_pair_redex_lemma,
psc_restriction_pair_lemma,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cat-comp_wf,
pi1_wf_top,
psc-restriction-comp,
subtype_rel_self,
iff_weakening_equal,
presheaf-type-ap-morph-comp,
cat-id_wf,
presheaf-type_wf,
small-category-subtype,
ps_context_wf,
small-category-cumulativity-2,
small-category_wf,
psc-restriction-id,
presheaf-type-ap-morph-id,
subtype_rel-equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
productElimination,
sqequalRule,
dependent_set_memberEquality_alt,
dependent_pairEquality_alt,
lambdaEquality_alt,
productEquality,
cumulativity,
hypothesis,
applyEquality,
because_Cache,
universeIsType,
productIsType,
inhabitedIsType,
functionIsType,
dependent_functionElimination,
Error :memTop,
independent_pairFormation,
lambdaFormation_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
independent_pairEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination,
equalityIstype,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[Gamma:ps\_context\{j:l\}(C)]. \mforall{}[A:\{Gamma \mvdash{}' \_\}].
(Gamma.A \mmember{} ps\_context\{[i | j]:l\}(C))
Date html generated:
2020_05_20-PM-01_27_11
Last ObjectModification:
2020_04_02-AM-11_41_05
Theory : presheaf!models!of!type!theory
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