Nuprl Lemma : pscm-presheaf-fun
∀C:SmallCategory. ∀X,Delta:ps_context{j:l}(C). ∀A,B:{X ⊢ _}. ∀s:psc_map{j:l}(C; Delta; X).
(((A ⟶ B))s = (Delta ⊢ (A)s ⟶ (B)s) ∈ {Delta ⊢ _})
Proof
Definitions occuring in Statement :
presheaf-fun: (A ⟶ B),
pscm-ap-type: (AF)s,
presheaf-type: {X ⊢ _},
psc_map: A ⟶ B,
ps_context: __⊢,
all: ∀x:A. B[x],
equal: s = t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
presheaf-type: {X ⊢ _},
uimplies: b supposing a,
pscm-ap-type: (AF)s,
presheaf-fun: (A ⟶ B),
presheaf-fun-family: presheaf-fun-family(C; X; A; B; I; a)
Lemmas referenced :
presheaf-type-equal,
pscm-ap-type_wf,
presheaf-fun_wf,
psc_map_wf,
small-category-cumulativity-2,
presheaf-type_wf,
ps_context_wf,
small-category_wf,
I_set_wf,
cat-ob_wf,
pscm-presheaf-fun-family,
presheaf-fun-family_wf,
pscm-ap_wf,
cat-arrow_wf,
presheaf-fun-family-comp,
psc-restriction_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
independent_isectElimination,
universeIsType,
instantiate,
because_Cache,
dependent_pairEquality_alt,
dependent_functionElimination,
functionIsType
Latex:
\mforall{}C:SmallCategory. \mforall{}X,Delta:ps\_context\{j:l\}(C). \mforall{}A,B:\{X \mvdash{} \_\}. \mforall{}s:psc\_map\{j:l\}(C; Delta; X).
(((A {}\mrightarrow{} B))s = (Delta \mvdash{} (A)s {}\mrightarrow{} (B)s))
Date html generated:
2020_05_20-PM-01_29_37
Last ObjectModification:
2020_04_02-PM-06_26_59
Theory : presheaf!models!of!type!theory
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