Nuprl Lemma : unit-functor-is-const
∀[A:SmallCategory]. ∀f:Functor(1;A). ∃a:cat-ob(A). (f = const-functor(A;a) ∈ Functor(1;A))
Proof
Definitions occuring in Statement :
const-functor: const-functor(A;a),
cat-functor: Functor(C1;C2),
unit-cat: 1,
cat-ob: cat-ob(C),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
unit: Unit,
cat-ob: cat-ob(C),
pi1: fst(t),
unit-cat: 1,
discrete-cat: discrete-cat(X),
mk-cat: mk-cat,
prop: ℙ,
uimplies: b supposing a,
cat-functor: Functor(C1;C2),
and: P ∧ Q,
top: Top,
const-functor: const-functor(A;a),
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
so_lambda: λ2x.t[x],
so_apply: x[s],
it: ⋅,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
functor-ob_wf,
unit-cat_wf,
it_wf,
subtype_rel_self,
equal-wf-base,
equal-functors,
const-functor_wf,
cat_ob_pair_lemma,
cat_arrow_triple_lemma,
cat_comp_tuple_lemma,
cat_id_tuple_lemma,
ob_pair_lemma,
ob_mk_functor_lemma,
cat-ob_wf,
arrow_pair_lemma,
arrow_mk_functor_lemma,
cat-arrow_wf,
equal_wf,
cat-functor_wf,
small-category_wf,
equal-unit,
squash_wf,
true_wf,
unit_wf2,
cat-id_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
dependent_pairFormation,
applyEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
hypothesisEquality,
sqequalRule,
intEquality,
baseClosed,
because_Cache,
independent_isectElimination,
setElimination,
rename,
productElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
equalityElimination,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
functionExtensionality,
natural_numberEquality,
imageMemberEquality,
independent_functionElimination
Latex:
\mforall{}[A:SmallCategory]. \mforall{}f:Functor(1;A). \mexists{}a:cat-ob(A). (f = const-functor(A;a))
Date html generated:
2020_05_20-AM-07_53_41
Last ObjectModification:
2017_07_28-AM-09_19_43
Theory : small!categories
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