### Nuprl Lemma : equal-functors

`∀[A,B:SmallCategory]. ∀[F,G:Functor(A;B)].`
`  (F = G ∈ Functor(A;B)) supposing `
`     ((∀x,y:cat-ob(A). ∀f:cat-arrow(A) x y.  ((F x y f) = (G x y f) ∈ (cat-arrow(B) (F x) (F y)))) and `
`     (∀x:cat-ob(A). ((F x) = (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]` all: `∀x:A. B[x]` apply: `f a` 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)` and: `P ∧ Q` all: `∀x:A. B[x]` top: `Top` 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]`
Lemmas referenced :  ob_pair_lemma arrow_pair_lemma equal_wf cat-ob_wf iff_weakening_equal cat-arrow_wf all_wf cat-id_wf cat-comp_wf functor-ob_wf functor-arrow_wf subtype_rel-equal squash_wf true_wf cat-functor_wf small-category_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution setElimination thin rename productElimination sqequalRule extract_by_obid dependent_functionElimination isect_memberEquality voidElimination voidEquality hypothesis dependent_set_memberEquality dependent_pairEquality functionExtensionality applyEquality lambdaEquality imageElimination isectElimination because_Cache hypothesisEquality natural_numberEquality imageMemberEquality baseClosed universeEquality equalityTransitivity equalitySymmetry independent_isectElimination independent_functionElimination functionEquality independent_pairFormation productEquality instantiate axiomEquality

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

Date html generated: 2020_05_20-AM-07_53_26
Last ObjectModification: 2017_07_28-AM-09_19_41

Theory : small!categories

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