Nuprl Lemma : presheaf-snd-pair
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[u:{X ⊢ _:A}]. ∀[v:{X ⊢ _:(B)[u]}].
(presheaf-pair(u;v).2 = v ∈ {X ⊢ _:(B)[u]})
Proof
Definitions occuring in Statement :
presheaf-pair: presheaf-pair(u;v),
presheaf-snd: p.2,
pscm-id-adjoin: [u],
psc-adjoin: X.A,
presheaf-term: {X ⊢ _:A},
pscm-ap-type: (AF)s,
presheaf-type: {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,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
presheaf-term: {X ⊢ _:A},
presheaf-pair: presheaf-pair(u;v),
presheaf-snd: p.2,
pi2: snd(t)
Lemmas referenced :
presheaf-term-equal,
presheaf-term_wf,
pscm-ap-type_wf,
psc-adjoin_wf,
small-category-cumulativity-2,
ps_context_cumulativity2,
presheaf-type-cumulativity2,
pscm-id-adjoin_wf,
presheaf-type_wf,
ps_context_wf,
small-category_wf,
I_set_wf,
cat-ob_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
equalitySymmetry,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
universeIsType,
instantiate,
applyEquality,
sqequalRule,
because_Cache,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
setElimination,
rename,
lambdaEquality_alt,
functionExtensionality
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[u:\{X \mvdash{} \_:A\}].
\mforall{}[v:\{X \mvdash{} \_:(B)[u]\}].
(presheaf-pair(u;v).2 = v)
Date html generated:
2020_05_20-PM-01_33_26
Last ObjectModification:
2020_04_02-PM-06_31_16
Theory : presheaf!models!of!type!theory
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