Nuprl Lemma : pscm-ap-id-type
∀[C:SmallCategory]. ∀[Gamma:ps_context{j:l}(C)]. ∀[A:{Gamma ⊢ _}]. ((A)1(Gamma) = A ∈ {Gamma ⊢ _})
Proof
Definitions occuring in Statement :
pscm-ap-type: (AF)s,
presheaf-type: {X ⊢ _},
pscm-id: 1(X),
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,
presheaf-type: {X ⊢ _},
and: P ∧ Q,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
ps_context: __⊢,
cat-functor: Functor(C1;C2),
pscm-id: 1(X),
pscm-ap-type: (AF)s,
pscm-ap: (s)x,
cat-ob: cat-ob(C),
pi1: fst(t),
type-cat: TypeCat
Lemmas referenced :
cat-ob_wf,
I_set_wf,
cat-id_wf,
subtype_rel-equal,
psc-restriction_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
psc-restriction-id,
small-category-cumulativity-2,
ps_context_cumulativity2,
subtype_rel_self,
iff_weakening_equal,
cat-arrow_wf,
cat-comp_wf,
psc-restriction-comp,
presheaf-type_wf,
ps_context_wf,
small-category_wf,
I_set_pair_redex_lemma,
psc_restriction_pair_lemma,
op-cat_wf,
cat_ob_op_lemma,
eta_conv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
equalitySymmetry,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality_alt,
hypothesis,
productElimination,
sqequalRule,
productIsType,
functionIsType,
universeIsType,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
equalityIstype,
because_Cache,
instantiate,
independent_isectElimination,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
Error :memTop,
dependent_pairEquality_alt,
functionExtensionality_alt
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[Gamma:ps\_context\{j:l\}(C)]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. ((A)1(Gamma) = A)
Date html generated:
2020_05_20-PM-01_26_20
Last ObjectModification:
2020_04_01-AM-11_00_56
Theory : presheaf!models!of!type!theory
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