Nuprl Lemma : functor-arrow_wf
∀[C,D:SmallCategory]. ∀[F:Functor(C;D)].
(functor-arrow(F) ∈ x:cat-ob(C)
⟶ y:cat-ob(C)
⟶ (cat-arrow(C) x y)
⟶ (cat-arrow(D) (functor-ob(F) x) (functor-ob(F) y)))
Proof
Definitions occuring in Statement :
functor-arrow: functor-arrow(F),
functor-ob: functor-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,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
pi1: fst(t),
functor-ob: functor-ob(F),
pi2: snd(t),
cat-functor: Functor(C1;C2),
functor-arrow: functor-arrow(F),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
small-category_wf,
cat-functor_wf
Rules used in proof :
because_Cache,
isect_memberEquality,
isectElimination,
lemma_by_obid,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
hypothesis,
hypothesisEquality,
productElimination,
rename,
thin,
setElimination,
sqequalHypSubstitution,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[C,D:SmallCategory]. \mforall{}[F:Functor(C;D)].
(functor-arrow(F) \mmember{} x:cat-ob(C)
{}\mrightarrow{} y:cat-ob(C)
{}\mrightarrow{} (cat-arrow(C) x y)
{}\mrightarrow{} (cat-arrow(D) (functor-ob(F) x) (functor-ob(F) y)))
Date html generated:
2016_05_18-AM-11_52_20
Last ObjectModification:
2015_12_28-PM-02_23_51
Theory : small!categories
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