Nuprl Lemma : mk-functor_wf
∀[A,B:SmallCategory]. ∀[F:cat-ob(A) ⟶ cat-ob(B)]. ∀[M:x:cat-ob(A)
⟶ y:cat-ob(A)
⟶ (cat-arrow(A) x y)
⟶ (cat-arrow(B) F[x] F[y])].
(functor(ob(a) = F[a];
arrow(x,y,f) = M[x;y;f]) ∈ Functor(A;B)) supposing
((∀x:cat-ob(A). (M[x;x;cat-id(A) x] = (cat-id(B) (F x)) ∈ (cat-arrow(B) (F x) (F x)))) and
(∀x,y,z:cat-ob(A). ∀f:cat-arrow(A) x y. ∀g:cat-arrow(A) y z.
(M[x;z;cat-comp(A) x y z f g]
= (cat-comp(B) (F x) (F y) (F z) M[x;y;f] M[y;z;g])
∈ (cat-arrow(B) (F x) (F z)))))
Proof
Definitions occuring in Statement :
mk-functor: mk-functor,
cat-functor: Functor(C1;C2),
cat-comp: cat-comp(C),
cat-id: cat-id(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2;s3],
so_apply: x[s],
all: ∀x:A. B[x],
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
cat-functor: Functor(C1;C2),
mk-functor: mk-functor,
so_apply: x[s],
so_apply: x[s1;s2;s3],
subtype_rel: A ⊆r B,
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
prop: ℙ
Lemmas referenced :
cat-arrow_wf,
all_wf,
equal_wf,
cat-id_wf,
cat-comp_wf,
cat-ob_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
dependent_set_memberEquality,
dependent_pairEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
because_Cache,
hypothesis,
sqequalHypSubstitution,
sqequalRule,
extract_by_obid,
isectElimination,
thin,
functionEquality,
independent_pairFormation,
productElimination,
productEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality
Latex:
\mforall{}[A,B:SmallCategory]. \mforall{}[F:cat-ob(A) {}\mrightarrow{} cat-ob(B)]. \mforall{}[M:x:cat-ob(A)
{}\mrightarrow{} y:cat-ob(A)
{}\mrightarrow{} (cat-arrow(A) x y)
{}\mrightarrow{} (cat-arrow(B) F[x] F[y])].
(functor(ob(a) = F[a];
arrow(x,y,f) = M[x;y;f]) \mmember{} Functor(A;B)) supposing
((\mforall{}x:cat-ob(A). (M[x;x;cat-id(A) x] = (cat-id(B) (F x)))) and
(\mforall{}x,y,z:cat-ob(A). \mforall{}f:cat-arrow(A) x y. \mforall{}g:cat-arrow(A) y z.
(M[x;z;cat-comp(A) x y z f g] = (cat-comp(B) (F x) (F y) (F z) M[x;y;f] M[y;z;g]))))
Date html generated:
2017_01_19-PM-02_52_12
Last ObjectModification:
2017_01_13-PM-00_33_53
Theory : small!categories
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