Nuprl Lemma : equal-functors1
∀[A,B:SmallCategory]. ∀[F:Functor(A;B)]. ∀[G:F:cat-ob(A) ⟶ cat-ob(B) × (x:cat-ob(A)
⟶ y:cat-ob(A)
⟶ (cat-arrow(A) x y)
⟶ (cat-arrow(B) (F x) (F y)))].
(F = G ∈ Functor(A;B)) supposing
((∀x,y:cat-ob(A). ∀f:cat-arrow(A) x y. ((F x y f) = ((snd(G)) x y f) ∈ (cat-arrow(B) (F x) (F y)))) and
(∀x:cat-ob(A). ((F x) = ((fst(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],
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
product: x:A × B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
cat-functor: Functor(C1;C2),
and: P ∧ Q,
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
pi2: snd(t),
pi1: fst(t),
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],
respects-equality: respects-equality(S;T)
Lemmas referenced :
ob_pair_lemma,
istype-void,
arrow_pair_lemma,
equal_wf,
cat-ob_wf,
subtype_rel_self,
iff_weakening_equal,
cat-arrow_wf,
cat-id_wf,
cat-comp_wf,
functor-ob_wf,
functor-arrow_wf,
pi2_wf,
subtype-respects-equality,
pi1_wf_top,
subtype_rel-equal,
subtype_rel_product,
top_wf,
cat-functor_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
productElimination,
sqequalRule,
introduction,
extract_by_obid,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
hypothesis,
dependent_set_memberEquality_alt,
dependent_pairEquality_alt,
functionExtensionality_alt,
applyEquality,
lambdaEquality_alt,
imageElimination,
isectElimination,
because_Cache,
hypothesisEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
independent_functionElimination,
universeIsType,
inhabitedIsType,
functionIsType,
independent_pairFormation,
productIsType,
equalityIstype,
functionEquality,
independent_pairEquality,
universeEquality,
lambdaFormation_alt
Latex:
\mforall{}[A,B:SmallCategory]. \mforall{}[F:Functor(A;B)]. \mforall{}[G:F:cat-ob(A) {}\mrightarrow{} cat-ob(B) \mtimes{} (x:cat-ob(A)
{}\mrightarrow{} y:cat-ob(A)
{}\mrightarrow{} (cat-arrow(A) x y)
{}\mrightarrow{} (cat-arrow(B) (F x)
(F y)))].
(F = G) supposing
((\mforall{}x,y:cat-ob(A). \mforall{}f:cat-arrow(A) x y. ((F x y f) = ((snd(G)) x y f))) and
(\mforall{}x:cat-ob(A). ((F x) = ((fst(G)) x))))
Date html generated:
2019_10_31-AM-07_24_10
Last ObjectModification:
2019_05_08-PM-01_27_09
Theory : small!categories
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