Nuprl Lemma : op-functor_wf
∀[C,D:SmallCategory]. ∀[F:Functor(C;D)]. (op-functor(F) ∈ Functor(op-cat(C);op-cat(D)))
Proof
Definitions occuring in Statement :
op-functor: op-functor(F),
op-cat: op-cat(C),
cat-functor: Functor(C1;C2),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
true: True,
prop: ℙ,
squash: ↓T,
so_apply: x[s1;s2;s3],
top: Top,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
op-functor: op-functor(F),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
op-cat-id,
op-cat-comp,
cat-id_wf,
functor-arrow-id,
iff_weakening_equal,
cat-comp_wf,
functor-arrow-comp,
true_wf,
squash_wf,
equal_wf,
small-category_wf,
cat-functor_wf,
cat-arrow_wf,
functor-arrow_wf,
op-cat-arrow,
cat-ob_wf,
subtype_rel-equal,
functor-ob_wf,
cat_ob_op_lemma,
op-cat_wf,
mk-functor_wf
Rules used in proof :
independent_functionElimination,
productElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
universeEquality,
imageElimination,
lambdaFormation,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
voidEquality,
voidElimination,
isect_memberEquality,
because_Cache,
independent_isectElimination,
applyEquality,
dependent_functionElimination,
lambdaEquality,
sqequalRule,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[C,D:SmallCategory]. \mforall{}[F:Functor(C;D)]. (op-functor(F) \mmember{} Functor(op-cat(C);op-cat(D)))
Date html generated:
2017_10_05-PM-03_34_04
Last ObjectModification:
2017_10_05-AM-11_32_49
Theory : small!categories
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