Nuprl Lemma : presheaf-fst-pair
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[u:{X ⊢ _:A}]. ∀[v:Top].
(presheaf-pair(u;v).1 = u ∈ {X ⊢ _:A})
Proof
Definitions occuring in Statement :
presheaf-pair: presheaf-pair(u;v),
presheaf-fst: p.1,
presheaf-term: {X ⊢ _:A},
presheaf-type: {X ⊢ _},
ps_context: __⊢,
uall: ∀[x:A]. B[x],
top: Top,
equal: s = t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
presheaf-fst: p.1,
pi1: fst(t),
presheaf-pair: presheaf-pair(u;v),
presheaf-term: {X ⊢ _:A},
uimplies: b supposing a,
subtype_rel: A ⊆r B
Lemmas referenced :
I_set_wf,
cat-ob_wf,
presheaf-term-equal,
istype-top,
presheaf-term_wf,
presheaf-type_wf,
ps_context_wf,
small-category-cumulativity-2,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
equalitySymmetry,
functionExtensionality,
sqequalRule,
hypothesis,
applyEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
hypothesisEquality,
extract_by_obid,
isectElimination,
equalityTransitivity,
independent_isectElimination,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
universeIsType,
instantiate
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[u:\{X \mvdash{} \_:A\}]. \mforall{}[v:Top].
(presheaf-pair(u;v).1 = u)
Date html generated:
2020_05_20-PM-01_33_30
Last ObjectModification:
2020_04_02-PM-06_31_19
Theory : presheaf!models!of!type!theory
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