Nuprl Lemma : functor-arrow-id
∀[C,D:SmallCategory]. ∀[F:Functor(C;D)].
∀x:cat-ob(C)
((functor-arrow(F) x x (cat-id(C) x))
= (cat-id(D) (functor-ob(F) x))
∈ (cat-arrow(D) (functor-ob(F) x) (functor-ob(F) x)))
Proof
Definitions occuring in Statement :
functor-arrow: functor-arrow(F),
functor-ob: functor-ob(F),
cat-functor: Functor(C1;C2),
cat-id: cat-id(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
top: Top,
mk-functor: mk-functor(ob;arrow),
and: P ∧ Q,
cat-functor: Functor(C1;C2),
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
small-category_wf,
cat-functor_wf,
cat-ob_wf,
functor_arrow_pair_lemma,
functor_ob_pair_lemma
Rules used in proof :
because_Cache,
axiomEquality,
lambdaEquality,
isectElimination,
hypothesisEquality,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
extract_by_obid,
sqequalRule,
productElimination,
rename,
thin,
setElimination,
sqequalHypSubstitution,
lambdaFormation,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[C,D:SmallCategory]. \mforall{}[F:Functor(C;D)].
\mforall{}x:cat-ob(C). ((functor-arrow(F) x x (cat-id(C) x)) = (cat-id(D) (functor-ob(F) x)))
Date html generated:
2017_01_11-AM-09_17_56
Last ObjectModification:
2017_01_10-PM-00_32_26
Theory : small!categories
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