Nuprl Lemma : sub-presheaf-set_wf
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[P:I:cat-ob(C) ⟶ X(I) ⟶ ℙ{j'}].
X | I,rho.P[I;rho] ∈ ps_context{j:l}(C) supposing psc-predicate(C; X; I,rho.P[I;rho])
Proof
Definitions occuring in Statement :
sub-presheaf-set: X | I,rho.P[I; rho],
psc-predicate: psc-predicate(C; X; I,rho.P[I; rho]),
I_set: A(I),
ps_context: __⊢,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
member: t ∈ T,
function: x:A ⟶ B[x],
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
sub-presheaf-set: X | I,rho.P[I; rho],
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
prop: ℙ,
so_apply: x[s1;s2],
presheaf: Presheaf(C),
cat-functor: Functor(C1;C2),
ps_context: __⊢,
I_set: A(I),
all: ∀x:A. B[x],
cat-ob: cat-ob(C),
pi1: fst(t),
type-cat: TypeCat,
guard: {T},
psc-predicate: psc-predicate(C; X; I,rho.P[I; rho])
Lemmas referenced :
psc-predicate_wf,
small-category-cumulativity-2,
I_set_wf,
cat-ob_wf,
ps_context_wf,
small-category_wf,
presheaf-subset_wf1,
subtype_rel_self,
functor-ob_wf,
op-cat_wf,
type-cat_wf,
subtype_rel-equal,
cat_ob_op_lemma,
subtype_rel_universe1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
thin,
instantiate,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
lambdaEquality_alt,
cumulativity,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
functionIsType,
universeEquality,
functionExtensionality,
functionEquality,
independent_isectElimination,
dependent_functionElimination,
because_Cache
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[P:I:cat-ob(C) {}\mrightarrow{} X(I) {}\mrightarrow{} \mBbbP{}\{j'\}].
X | I,rho.P[I;rho] \mmember{} ps\_context\{j:l\}(C) supposing psc-predicate(C; X; I,rho.P[I;rho])
Date html generated:
2020_05_20-PM-01_23_28
Last ObjectModification:
2020_04_03-AM-11_46_06
Theory : presheaf!models!of!type!theory
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