Nuprl Lemma : psdcff-inj_wf
∀[C:SmallCategory]. ∀[A,B:Type]. ∀[X:ps_context{j:l}(C)]. ∀[I:cat-ob(C)]. ∀[a:X(I)]. ∀[w:presheaf-fun-family(C;
X;
discr(A);
discr(B);
I;
a)].
(psdcff-inj(I;w) ∈ A ⟶ B)
Proof
Definitions occuring in Statement :
psdcff-inj: psdcff-inj(I;w),
discrete-presheaf-type: discr(T),
presheaf-fun-family: presheaf-fun-family(C; X; A; B; I; a),
I_set: A(I),
ps_context: __⊢,
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type,
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
psdcff-inj: psdcff-inj(I;w),
presheaf-fun-family: presheaf-fun-family(C; X; A; B; I; a),
subtype_rel: A ⊆r B,
presheaf-type-at: A(a),
pi1: fst(t),
discrete-presheaf-type: discr(T)
Lemmas referenced :
cat-id_wf,
subtype_rel_self,
presheaf-fun-family_wf,
discrete-presheaf-type_wf,
I_set_wf,
cat-ob_wf,
ps_context_wf,
small-category-cumulativity-2,
istype-universe,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
applyEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
hypothesisEquality,
hypothesis,
extract_by_obid,
isectElimination,
functionEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
universeEquality
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[A,B:Type]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[I:cat-ob(C)]. \mforall{}[a:X(I)].
\mforall{}[w:presheaf-fun-family(C; X; discr(A); discr(B); I; a)].
(psdcff-inj(I;w) \mmember{} A {}\mrightarrow{} B)
Date html generated:
2020_05_20-PM-01_35_47
Last ObjectModification:
2020_04_02-PM-06_35_51
Theory : presheaf!models!of!type!theory
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