Nuprl Lemma : ps-context-map_wf
∀C:SmallCategory. ∀I:cat-ob(C). ∀Gamma:ps_context{j:l}(C). ∀rho:Gamma(I). (<rho> ∈ psc_map{j:l}(C; Yoneda(I); Gamma))
Proof
Definitions occuring in Statement :
ps-context-map: <rho>,
psc_map: A ⟶ B,
Yoneda: Yoneda(I),
I_set: A(I),
ps_context: __⊢,
all: ∀x:A. B[x],
member: t ∈ T,
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
ps-context-map: <rho>,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
ps_context: __⊢,
cat-functor: Functor(C1;C2),
and: P ∧ Q,
small-category: SmallCategory,
spreadn: spread4,
type-cat: TypeCat,
op-cat: op-cat(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
pi2: snd(t),
pi1: fst(t),
Yoneda: Yoneda(I),
functor-ob: ob(F),
compose: f o g,
I_set: A(I),
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
cat_comp_tuple_lemma,
cat_id_tuple_lemma,
cat_arrow_triple_lemma,
arrow_pair_lemma,
ob_pair_lemma,
equal_wf,
squash_wf,
true_wf,
istype-universe,
subtype_rel_self,
iff_weakening_equal,
compose_wf,
I_set_wf,
ps_context_wf,
small-category-cumulativity-2,
cat-ob_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalRule,
hypothesisEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
introduction,
extract_by_obid,
dependent_functionElimination,
Error :memTop,
hypothesis,
dependent_set_memberEquality_alt,
lambdaEquality_alt,
applyEquality,
universeIsType,
inhabitedIsType,
functionExtensionality,
instantiate,
imageElimination,
isectElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination,
functionIsType,
equalityIstype
Latex:
\mforall{}C:SmallCategory. \mforall{}I:cat-ob(C). \mforall{}Gamma:ps\_context\{j:l\}(C). \mforall{}rho:Gamma(I).
(<rho> \mmember{} psc\_map\{j:l\}(C; Yoneda(I); Gamma))
Date html generated:
2020_05_20-PM-01_23_54
Last ObjectModification:
2020_04_01-AM-10_47_21
Theory : presheaf!models!of!type!theory
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