Nuprl Lemma : presheaf-type-ap-morph_wf
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[I,J:cat-ob(C)]. ∀[f:cat-arrow(C) J I]. ∀[a:X(I)].
∀[u:A(a)].
((u a f) ∈ A(f(a)))
Proof
Definitions occuring in Statement :
presheaf-type-ap-morph: (u a f),
presheaf-type-at: A(a),
presheaf-type: {X ⊢ _},
psc-restriction: f(s),
I_set: A(I),
ps_context: __⊢,
uall: ∀[x:A]. B[x],
member: t ∈ T,
apply: f a,
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
presheaf-type-ap-morph: (u a f),
presheaf-type: {X ⊢ _},
presheaf-type-at: A(a),
pi1: fst(t),
pi2: snd(t),
subtype_rel: A ⊆r B
Lemmas referenced :
presheaf-type-at_wf,
I_set_wf,
cat-arrow_wf,
cat-ob_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,
sqequalRule,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
applyEquality,
hypothesisEquality,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
extract_by_obid,
isectElimination,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
instantiate
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[I,J:cat-ob(C)]. \mforall{}[f:cat-arrow(C) J I].
\mforall{}[a:X(I)]. \mforall{}[u:A(a)].
((u a f) \mmember{} A(f(a)))
Date html generated:
2020_05_20-PM-01_25_54
Last ObjectModification:
2020_04_01-AM-11_51_03
Theory : presheaf!models!of!type!theory
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