Nuprl Lemma : pscm-id-adjoin-wf
∀[C:SmallCategory]. ∀[Gamma:ps_context{j:l}(C)]. ∀[A:{Gamma ⊢' _}]. ∀[u:{Gamma ⊢ _:A}].
([u] ∈ psc_map{[i | j]:l}(C; Gamma; Gamma.A))
Proof
Definitions occuring in Statement :
pscm-id-adjoin: [u],
psc-adjoin: X.A,
presheaf-term: {X ⊢ _:A},
presheaf-type: {X ⊢ _},
psc_map: A ⟶ B,
ps_context: __⊢,
uall: ∀[x:A]. B[x],
member: t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
pscm-id-adjoin: [u],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
presheaf-term_wf,
presheaf-type_wf,
small-category-subtype,
ps_context_wf,
small-category-cumulativity-2,
small-category_wf,
ps_context_cumulativity2,
presheaf-type-cumulativity2,
pscm-id_wf,
subtype_rel-equal,
pscm-ap-type_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
pscm-ap-id-type,
subtype_rel_self,
iff_weakening_equal,
pscm-adjoin-wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
universeIsType,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
independent_isectElimination,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[Gamma:ps\_context\{j:l\}(C)]. \mforall{}[A:\{Gamma \mvdash{}' \_\}]. \mforall{}[u:\{Gamma \mvdash{} \_:A\}].
([u] \mmember{} psc\_map\{[i | j]:l\}(C; Gamma; Gamma.A))
Date html generated:
2020_05_20-PM-01_27_59
Last ObjectModification:
2020_04_02-PM-01_51_53
Theory : presheaf!models!of!type!theory
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