Nuprl Lemma : psc-fst-pscm-adjoin
∀[C:SmallCategory]. ∀[Gamma,Delta:ps_context{j:l}(C)]. ∀[A:{Gamma ⊢ _}]. ∀[sigma:psc_map{j:l}(C; Delta; Gamma)].
∀[u:{Delta ⊢ _:(A)sigma}].
(p o (sigma;u) = sigma ∈ psc_map{j:l}(C; Delta; Gamma))
Proof
Definitions occuring in Statement :
pscm-adjoin: (s;u),
psc-fst: p,
psc-adjoin: X.A,
presheaf-term: {X ⊢ _:A},
pscm-ap-type: (AF)s,
presheaf-type: {X ⊢ _},
pscm-comp: G o F,
psc_map: A ⟶ B,
ps_context: __⊢,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
subtype_rel: A ⊆r B,
psc-fst: p,
pscm-adjoin: (s;u),
pscm-comp: G o F,
functor-ob: ob(F),
type-cat: TypeCat,
cat-arrow: cat-arrow(C),
pi2: snd(t),
pi1: fst(t),
pscm-ap: (s)x,
compose: f o g,
ps_context: __⊢,
cat-functor: Functor(C1;C2),
all: ∀x:A. B[x],
and: P ∧ Q,
cat-ob: cat-ob(C),
small-category: SmallCategory,
spreadn: spread4,
op-cat: op-cat(C),
functor-arrow: arrow(F),
cat-comp: cat-comp(C)
Lemmas referenced :
presheaf-term_wf,
pscm-ap-type_wf,
psc_map_wf,
small-category-cumulativity-2,
presheaf-type_wf,
ps_context_wf,
small-category_wf,
cat-ob_wf,
op-cat_wf,
ob_pair_lemma,
arrow_pair_lemma,
subtype_rel_self,
cat_id_tuple_lemma,
compose_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
equalitySymmetry,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality_alt,
hypothesis,
universeIsType,
extract_by_obid,
isectElimination,
hypothesisEquality,
sqequalRule,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
applyEquality,
because_Cache,
functionExtensionality_alt,
productElimination,
dependent_functionElimination,
Error :memTop,
universeEquality,
functionIsType,
equalityIstype
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[Gamma,Delta:ps\_context\{j:l\}(C)]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[sigma:psc\_map\{j:l\}(C;
Delta;
Gamma
)].
\mforall{}[u:\{Delta \mvdash{} \_:(A)sigma\}].
(p o (sigma;u) = sigma)
Date html generated:
2020_05_20-PM-01_28_10
Last ObjectModification:
2020_04_02-PM-01_42_27
Theory : presheaf!models!of!type!theory
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