### 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: 2020_05_20-AM-07_50_47
Last ObjectModification: 2017_01_13-PM-00_33_53

Theory : small!categories

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