Nuprl Lemma : sub-presheaf-set_functionality
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[P,Q:I:cat-ob(C) ⟶ X(I) ⟶ ℙ].
X | I,rho.P[I;rho] ≡ X | I,rho.Q[I;rho]
supposing (∀I:cat-ob(C). ∀rho:X(I). (P[I;rho] ⇐⇒ Q[I;rho])) ∧ 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),
ext-eq-psc: X ≡ Y,
ps_context: __⊢,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
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,
and: P ∧ Q,
ext-eq-psc: X ≡ Y,
ext-equal-presheaves: ext-equal-presheaves(C;F;G),
sub-presheaf-set: X | I,rho.P[I; rho],
presheaf-subset: F|I,rho.P[I; rho],
mk-presheaf: mk-presheaf,
all: ∀x:A. B[x],
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
ps_context: __⊢,
so_apply: x[s1;s2],
prop: ℙ,
I_set: A(I),
ext-eq: A ≡ B,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x y.t[x; y],
guard: {T},
psc-predicate: psc-predicate(C; X; I,rho.P[I; rho]),
stable-element-predicate: stable-element-predicate(C;F;I,rho.P[I; rho])
Lemmas referenced :
ob_mk_functor_lemma,
arrow_mk_functor_lemma,
functor-arrow_wf,
op-cat_wf,
type-cat_wf,
subtype_rel-equal,
cat-ob_wf,
cat_ob_op_lemma,
cat-arrow_wf,
op-cat-arrow,
functor-ob_wf,
I_set_wf,
psc-predicate_wf,
small-category-cumulativity-2,
ps_context_cumulativity2,
ps_context_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
sqequalRule,
extract_by_obid,
dependent_functionElimination,
Error :memTop,
hypothesis,
lambdaFormation_alt,
because_Cache,
lambdaEquality_alt,
setElimination,
rename,
dependent_set_memberEquality_alt,
applyEquality,
instantiate,
isectElimination,
hypothesisEquality,
independent_isectElimination,
universeIsType,
setIsType,
independent_pairEquality,
axiomEquality,
functionIsTypeImplies,
inhabitedIsType,
productIsType,
functionIsType,
cumulativity,
isect_memberEquality_alt,
isectIsTypeImplies,
universeEquality,
independent_functionElimination
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[P,Q:I:cat-ob(C) {}\mrightarrow{} X(I) {}\mrightarrow{} \mBbbP{}].
X | I,rho.P[I;rho] \mequiv{} X | I,rho.Q[I;rho]
supposing (\mforall{}I:cat-ob(C). \mforall{}rho:X(I). (P[I;rho] \mLeftarrow{}{}\mRightarrow{} Q[I;rho]))
\mwedge{} psc-predicate(C; X; I,rho.P[I;rho])
Date html generated:
2020_05_20-PM-01_23_31
Last ObjectModification:
2020_04_01-AM-10_45_19
Theory : presheaf!models!of!type!theory
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