Nuprl Lemma : psc-snd_wf
∀[C:SmallCategory]. ∀[Gamma:ps_context{j:l}(C)]. ∀[A:{Gamma ⊢ _}]. (q ∈ {Gamma.A ⊢ _:(A)p})
Proof
Definitions occuring in Statement :
psc-snd: q,
psc-fst: p,
psc-adjoin: X.A,
presheaf-term: {X ⊢ _:A},
pscm-ap-type: (AF)s,
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,
presheaf-term: {X ⊢ _:A},
subtype_rel: A ⊆r B,
presheaf-type: {X ⊢ _},
ps_context: __⊢,
cat-functor: Functor(C1;C2),
psc-snd: q,
psc-fst: p,
pscm-ap-type: (AF)s,
presheaf-type-at: A(a),
psc-adjoin: X.A,
I_set: A(I),
all: ∀x:A. B[x],
pi1: fst(t),
pscm-ap: (s)x,
pi2: snd(t),
psc-restriction: f(s),
uimplies: b supposing a,
cat-ob: cat-ob(C),
type-cat: TypeCat,
presheaf-type-ap-morph: (u a f),
functor-ob: ob(F),
and: P ∧ Q,
implies: P ⇒ Q,
respects-equality: respects-equality(S;T)
Lemmas referenced :
presheaf-type_wf,
ps_context_wf,
small-category-cumulativity-2,
small-category_wf,
ob_pair_lemma,
I_set_pair_redex_lemma,
presheaf_type_at_pair_lemma,
subtype_rel-equal,
cat-ob_wf,
op-cat_wf,
cat_ob_op_lemma,
subtype_rel_self,
pscm-ap-type-at,
psc_restriction_pair_lemma,
presheaf_type_ap_morph_pair_lemma,
I_set_wf,
psc-adjoin_wf,
cat-arrow_wf,
subtype-respects-equality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
dependent_set_memberEquality_alt,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
applyEquality,
setElimination,
rename,
productElimination,
dependent_functionElimination,
Error :memTop,
lambdaEquality_alt,
productIsType,
independent_isectElimination,
universeEquality,
lambdaFormation_alt,
functionIsType,
because_Cache,
equalityIstype,
dependent_pairEquality_alt,
independent_pairEquality,
independent_functionElimination
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[Gamma:ps\_context\{j:l\}(C)]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. (q \mmember{} \{Gamma.A \mvdash{} \_:(A)p\})
Date html generated:
2020_05_20-PM-01_27_27
Last ObjectModification:
2020_04_02-AM-11_21_50
Theory : presheaf!models!of!type!theory
Home
Index