Nuprl Lemma : poset-extend-unique
∀[C:SmallCategory]. ∀[I:Cname List]. ∀[L:name-morph(I;[]) ⟶ cat-ob(C)]. ∀[E:i:nameset(I)
⟶ c:{c:name-morph(I;[])| (c i) = 0 ∈ ℕ2}
⟶ (cat-arrow(C) (L c) (L flip(c;i)))].
∀[F,G:Functor(poset-cat(I);C)].
(F = G ∈ Functor(poset-cat(I);C)) supposing (poset-functor-extends(C;I;L;E;G) and poset-functor-extends(C;I;L;E;F))
Proof
Definitions occuring in Statement :
poset-functor-extends: poset-functor-extends(C;I;L;E;F),
poset-cat: poset-cat(J),
name-morph-flip: flip(f;y),
name-morph: name-morph(I;J),
nameset: nameset(L),
coordinate_name: Cname,
cat-functor: Functor(C1;C2),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
nil: [],
list: T List,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
name-morph: name-morph(I;J),
poset-cat: poset-cat(J),
pi1: fst(t),
cat-ob: cat-ob(C),
subtype_rel: A ⊆r B,
prop: ℙ,
and: P ∧ Q,
top: Top,
all: ∀x:A. B[x],
poset-functor-extends: poset-functor-extends(C;I;L;E;F),
cat-functor: Functor(C1;C2),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
not: ¬A,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
nat: ℕ,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
true: True,
squash: ↓T,
cand: A c∧ B,
ge: i ≥ j ,
uiff: uiff(P;Q)
Lemmas referenced :
small-category_wf,
list_wf,
name-morph-flip_wf,
extd-nameset-nil,
int_seg_wf,
equal-wf-T-base,
cat-functor_wf,
nameset_wf,
poset-functor-extends_wf,
cat-comp_wf,
cat-id_wf,
all_wf,
cat-arrow_wf,
poset-cat_wf,
subtype_rel_self,
cat-ob_wf,
coordinate_name_wf,
nil_wf,
name-morph_wf,
equal_wf,
and_wf,
arrow_pair_lemma,
ob_pair_lemma,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
itermVar_wf,
intformle_wf,
intformnot_wf,
satisfiable-full-omega-tt,
nat_wf,
decidable__le,
poset-cat-dist_wf,
iff_weakening_equal,
true_wf,
squash_wf,
poset-cat-arrow-unique,
subtype_rel-equal,
subtype_rel_dep_function,
istype-universe,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
le_wf,
less_than_wf,
ge_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermConstant_wf,
intformand_wf,
nat_properties,
poset-cat-dist-zero,
poset-cat-arrow-cases,
equal_functionality_wrt_subtype_rel2,
poset-cat-arrow-equals,
poset-cat-dist-add,
false_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
itermAdd_wf,
intformeq_wf,
add-is-int-iff
Rules used in proof :
baseClosed,
natural_numberEquality,
setEquality,
axiomEquality,
lambdaEquality,
productEquality,
because_Cache,
applyEquality,
applyLambdaEquality,
hypothesisEquality,
functionEquality,
isectElimination,
independent_pairFormation,
equalitySymmetry,
equalityTransitivity,
functionExtensionality,
dependent_pairEquality,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
extract_by_obid,
sqequalRule,
productElimination,
dependent_set_memberEquality,
rename,
thin,
setElimination,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
computeAll,
intEquality,
int_eqEquality,
dependent_pairFormation,
independent_isectElimination,
unionElimination,
independent_functionElimination,
hyp_replacement,
imageMemberEquality,
universeEquality,
imageElimination,
lambdaFormation,
dependent_set_memberEquality_alt,
productIsType,
equalityIsType1,
inhabitedIsType,
lambdaEquality_alt,
instantiate,
universeIsType,
lambdaFormation_alt,
intWeakElimination,
cumulativity,
comment,
closedConclusion,
baseApply,
pointwiseFunctionality,
promote_hyp
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[I:Cname List]. \mforall{}[L:name-morph(I;[]) {}\mrightarrow{} cat-ob(C)].
\mforall{}[E:i:nameset(I) {}\mrightarrow{} c:\{c:name-morph(I;[])| (c i) = 0\} {}\mrightarrow{} (cat-arrow(C) (L c) (L flip(c;i)))].
\mforall{}[F,G:Functor(poset-cat(I);C)].
(F = G) supposing (poset-functor-extends(C;I;L;E;G) and poset-functor-extends(C;I;L;E;F))
Date html generated:
2019_11_05-PM-00_34_56
Last ObjectModification:
2018_11_07-AM-11_39_20
Theory : cubical!sets
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