Nuprl Lemma : equal-functors
∀[A,B:SmallCategory]. ∀[F,G:Functor(A;B)].
(F = G ∈ Functor(A;B)) supposing
((∀x,y:cat-ob(A). ∀f:cat-arrow(A) x y.
((arrow(F) x y f) = (arrow(G) x y f) ∈ (cat-arrow(B) (ob(F) x) (ob(F) y)))) and
(∀x:cat-ob(A). ((ob(F) x) = (ob(G) x) ∈ cat-ob(B))))
Proof
Definitions occuring in Statement :
functor-arrow: arrow(F),
functor-ob: ob(F),
cat-functor: Functor(C1;C2),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
cat-functor: Functor(C1;C2),
and: P ∧ Q,
all: ∀x:A. B[x],
top: Top,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
ob_pair_lemma,
arrow_pair_lemma,
equal_wf,
cat-ob_wf,
iff_weakening_equal,
cat-arrow_wf,
all_wf,
cat-id_wf,
cat-comp_wf,
functor-ob_wf,
functor-arrow_wf,
subtype_rel-equal,
squash_wf,
true_wf,
cat-functor_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
sqequalRule,
extract_by_obid,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
dependent_set_memberEquality,
dependent_pairEquality,
functionExtensionality,
applyEquality,
lambdaEquality,
imageElimination,
isectElimination,
because_Cache,
hypothesisEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
independent_functionElimination,
functionEquality,
independent_pairFormation,
productEquality,
instantiate,
axiomEquality
Latex:
\mforall{}[A,B:SmallCategory]. \mforall{}[F,G:Functor(A;B)].
(F = G) supposing
((\mforall{}x,y:cat-ob(A). \mforall{}f:cat-arrow(A) x y. ((arrow(F) x y f) = (arrow(G) x y f))) and
(\mforall{}x:cat-ob(A). ((ob(F) x) = (ob(G) x))))
Date html generated:
2017_10_05-AM-00_47_26
Last ObjectModification:
2017_07_28-AM-09_19_41
Theory : small!categories
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