Nuprl Lemma : functor_arrow_wf
∀[C,D:SmallCategory]. ∀[F:Functor(C;D)]. ∀[x,y:cat-ob(C)]. ∀[f:cat-arrow(C) x y]. (F(f) ∈ cat-arrow(D) (F x) (F y))
Proof
Definitions occuring in Statement :
functor_arrow: F(f),
functor-ob: ob(F),
cat-functor: Functor(C1;C2),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
member: t ∈ T,
apply: f a
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
functor_arrow: F(f)
Lemmas referenced :
functor-arrow_wf,
cat-arrow_wf,
cat-ob_wf,
cat-functor_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
applyEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
because_Cache
Latex:
\mforall{}[C,D:SmallCategory]. \mforall{}[F:Functor(C;D)]. \mforall{}[x,y:cat-ob(C)]. \mforall{}[f:cat-arrow(C) x y].
(F(f) \mmember{} cat-arrow(D) (F x) (F y))
Date html generated:
2020_05_20-AM-07_51_01
Last ObjectModification:
2017_01_17-PM-00_44_46
Theory : small!categories
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