Nuprl Lemma : psc-map-subtype
∀[C:SmallCategory]. ∀[A,B:ps_context{j:l}(C)]. (psc_map{j:l}(C; A; B) ⊆r (I:cat-ob(C) ⟶ A(I) ⟶ B(I)))
Proof
Definitions occuring in Statement :
psc_map: A ⟶ B,
I_set: A(I),
ps_context: __⊢,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
member: t ∈ T,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
small-category: SmallCategory,
spreadn: spread4,
and: P ∧ Q,
I_set: A(I),
type-cat: TypeCat,
cat-arrow: cat-arrow(C),
op-cat: op-cat(C),
cat-ob: cat-ob(C),
pi2: snd(t),
pi1: fst(t),
all: ∀x:A. B[x]
Lemmas referenced :
psc_map_wf,
ps_context_wf,
small-category-cumulativity-2,
small-category_wf,
cat_ob_pair_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaEquality_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
hypothesis,
universeIsType,
instantiate,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
sqequalRule,
productElimination,
dependent_functionElimination,
Error :memTop
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[A,B:ps\_context\{j:l\}(C)].
(psc\_map\{j:l\}(C; A; B) \msubseteq{}r (I:cat-ob(C) {}\mrightarrow{} A(I) {}\mrightarrow{} B(I)))
Date html generated:
2020_05_20-PM-01_24_01
Last ObjectModification:
2020_04_01-AM-10_47_05
Theory : presheaf!models!of!type!theory
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