Nuprl Lemma : functor-is-isomorphism
∀[A,B:SmallCategory].
∀f:Functor(A;B)
(cat-isomorphism(Cat;A;B;f)
⇐⇒ Bij(cat-ob(A);cat-ob(B);ob(f)) ∧ (∀x,y:cat-ob(A). Bij(cat-arrow(A) x y;cat-arrow(B) (f x) (f y);f x y)))
Proof
Definitions occuring in Statement :
cat-cat: Cat,
functor-arrow: arrow(F),
functor-ob: ob(F),
cat-functor: Functor(C1;C2),
cat-isomorphism: cat-isomorphism(C;x;y;f),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
biject: Bij(A;B;f),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
apply: f a
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
cat-cat: Cat,
cat-isomorphism: cat-isomorphism(C;x;y;f),
member: t ∈ T,
top: Top,
cat-inverse: fg=1,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
id_functor: 1,
functor-comp: functor-comp(F;G),
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
biject: Bij(A;B;f),
inject: Inj(A;B;f),
surject: Surj(A;B;f),
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
cand: A c∧ B,
pi1: fst(t)
Lemmas referenced :
cat_arrow_triple_lemma,
cat_comp_tuple_lemma,
cat_id_tuple_lemma,
exists_wf,
cat-functor_wf,
equal-wf-T-base,
functor-comp_wf,
biject_wf,
cat-ob_wf,
functor-ob_wf,
all_wf,
cat-arrow_wf,
functor-arrow_wf,
small-category_wf,
ob_mk_functor_lemma,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal,
arrow_mk_functor_lemma,
subtype_rel-equal,
biject-inverse,
small-category-subtype,
subtype_rel_dep_function,
mk-functor_wf,
functor-arrow-comp,
cat-comp_wf,
functor-arrow-id,
cat-id_wf,
id_functor_wf,
equal-functors
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
independent_pairFormation,
isectElimination,
hypothesisEquality,
lambdaEquality,
productEquality,
baseClosed,
because_Cache,
applyEquality,
productElimination,
applyLambdaEquality,
hyp_replacement,
equalitySymmetry,
imageElimination,
equalityTransitivity,
universeEquality,
natural_numberEquality,
imageMemberEquality,
independent_isectElimination,
independent_functionElimination,
dependent_pairFormation,
dependent_set_memberEquality,
setElimination,
rename,
instantiate,
functionExtensionality,
functionEquality,
cumulativity,
promote_hyp
Latex:
\mforall{}[A,B:SmallCategory].
\mforall{}f:Functor(A;B)
(cat-isomorphism(Cat;A;B;f)
\mLeftarrow{}{}\mRightarrow{} Bij(cat-ob(A);cat-ob(B);ob(f))
\mwedge{} (\mforall{}x,y:cat-ob(A). Bij(cat-arrow(A) x y;cat-arrow(B) (f x) (f y);f x y)))
Date html generated:
2020_05_20-AM-07_53_56
Last ObjectModification:
2017_07_28-AM-09_19_47
Theory : small!categories
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