### Nuprl Lemma : equal-functors1

`∀[A,B:SmallCategory]. ∀[F:Functor(A;B)]. ∀[G:F:cat-ob(A) ⟶ cat-ob(B) × (x:cat-ob(A)`
`                                                                        ⟶ y:cat-ob(A)`
`                                                                        ⟶ (cat-arrow(A) x y)`
`                                                                        ⟶ (cat-arrow(B) (F x) (F y)))].`
`  (F = G ∈ Functor(A;B)) supposing `
`     ((∀x,y:cat-ob(A). ∀f:cat-arrow(A) x y.  ((F x y f) = ((snd(G)) x y f) ∈ (cat-arrow(B) (F x) (F y)))) and `
`     (∀x:cat-ob(A). ((F x) = ((fst(G)) x) ∈ cat-ob(B))))`

Proof

Definitions occuring in Statement :  functor-arrow: `arrow(F)` functor-ob: `ob(F)` cat-functor: `Functor(C1;C2)` cat-arrow: `cat-arrow(C)` cat-ob: `cat-ob(C)` small-category: `SmallCategory` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` pi1: `fst(t)` pi2: `snd(t)` all: `∀x:A. B[x]` apply: `f a` function: `x:A ⟶ B[x]` product: `x:A × B[x]` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` uimplies: `b supposing a` cat-functor: `Functor(C1;C2)` and: `P ∧ Q` all: `∀x:A. B[x]` member: `t ∈ T` top: `Top` pi2: `snd(t)` pi1: `fst(t)` squash: `↓T` true: `True` subtype_rel: `A ⊆r B` prop: `ℙ` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` implies: `P `` Q` cand: `A c∧ B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` respects-equality: `respects-equality(S;T)`
Lemmas referenced :  ob_pair_lemma istype-void arrow_pair_lemma equal_wf cat-ob_wf subtype_rel_self iff_weakening_equal cat-arrow_wf cat-id_wf cat-comp_wf functor-ob_wf functor-arrow_wf pi2_wf subtype-respects-equality pi1_wf_top subtype_rel-equal subtype_rel_product top_wf cat-functor_wf small-category_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt sqequalHypSubstitution setElimination thin rename cut productElimination sqequalRule introduction extract_by_obid dependent_functionElimination isect_memberEquality_alt voidElimination hypothesis dependent_set_memberEquality_alt dependent_pairEquality_alt functionExtensionality_alt applyEquality lambdaEquality_alt imageElimination isectElimination because_Cache hypothesisEquality natural_numberEquality imageMemberEquality baseClosed instantiate equalityTransitivity equalitySymmetry independent_isectElimination independent_functionElimination universeIsType inhabitedIsType functionIsType independent_pairFormation productIsType equalityIstype functionEquality independent_pairEquality universeEquality lambdaFormation_alt

Latex:
\mforall{}[A,B:SmallCategory].  \mforall{}[F:Functor(A;B)].  \mforall{}[G:F:cat-ob(A)  {}\mrightarrow{}  cat-ob(B)  \mtimes{}  (x:cat-ob(A)
{}\mrightarrow{}  y:cat-ob(A)
{}\mrightarrow{}  (cat-arrow(A)  x  y)
{}\mrightarrow{}  (cat-arrow(B)  (F  x)
(F  y)))].
(F  =  G)  supposing
((\mforall{}x,y:cat-ob(A).  \mforall{}f:cat-arrow(A)  x  y.    ((F  x  y  f)  =  ((snd(G))  x  y  f)))  and
(\mforall{}x:cat-ob(A).  ((F  x)  =  ((fst(G))  x))))

Date html generated: 2020_05_20-AM-07_53_24
Last ObjectModification: 2019_05_08-PM-01_27_09

Theory : small!categories

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