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
Home
Index