Nuprl Lemma : psc-map-is
∀[C:SmallCategory]. ∀[A,B:ps_context{j:l}(C)].
(psc_map{j:l}(C; A; B) ~ {trans:I:cat-ob(C) ⟶ A(I) ⟶ B(I)|
∀I,J:cat-ob(C). ∀g:cat-arrow(C) J I.
((λs.g(trans I s)) = (λs.(trans J g(s))) ∈ (A(I) ⟶ B(J)))} )
Proof
Definitions occuring in Statement :
psc_map: A ⟶ B,
psc-restriction: f(s),
I_set: A(I),
ps_context: __⊢,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
sqequal: s ~ t,
equal: s = t ∈ T,
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,
small-category: SmallCategory,
spreadn: spread4,
psc-restriction: f(s),
I_set: A(I),
psc_map: A ⟶ B,
all: ∀x:A. B[x],
type-cat: TypeCat,
op-cat: op-cat(C),
nat-trans: nat-trans(C;D;F;G),
functor-arrow: arrow(F),
compose: f o g,
subtype_rel: A ⊆r B
Lemmas referenced :
cat_ob_pair_lemma,
cat_arrow_triple_lemma,
cat_comp_tuple_lemma,
ps_context_wf,
small-category-cumulativity-2,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
sqequalRule,
extract_by_obid,
dependent_functionElimination,
Error :memTop,
hypothesis,
axiomSqEquality,
inhabitedIsType,
hypothesisEquality,
isect_memberEquality_alt,
isectElimination,
isectIsTypeImplies,
universeIsType,
instantiate,
applyEquality
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[A,B:ps\_context\{j:l\}(C)].
(psc\_map\{j:l\}(C; A; B) \msim{} \{trans:I:cat-ob(C) {}\mrightarrow{} A(I) {}\mrightarrow{} B(I)|
\mforall{}I,J:cat-ob(C). \mforall{}g:cat-arrow(C) J I.
((\mlambda{}s.g(trans I s)) = (\mlambda{}s.(trans J g(s))))\} )
Date html generated:
2020_05_20-PM-01_24_35
Last ObjectModification:
2020_04_01-AM-11_00_39
Theory : presheaf!models!of!type!theory
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