Nuprl Lemma : rep-sub-sheaf_wf
∀[C:SmallCategory]. ∀[X:cat-ob(C)]. ∀[P:U:cat-ob(C) ⟶ (cat-arrow(C) U X) ⟶ ℙ].
rep-sub-sheaf(C;X;P) ∈ Functor(op-cat(C);TypeCat)
supposing ∀A,B:cat-ob(C). ∀g:cat-arrow(C) A B. ∀b:{b:cat-arrow(C) B X| P B b} . (P A (cat-comp(C) A B X g b))
Proof
Definitions occuring in Statement :
rep-sub-sheaf: rep-sub-sheaf(C;X;P),
type-cat: TypeCat,
op-cat: op-cat(C),
cat-functor: Functor(C1;C2),
cat-comp: cat-comp(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
rep-sub-sheaf: rep-sub-sheaf(C;X;P),
cat-functor: Functor(C1;C2),
small-category: SmallCategory,
cat-comp: cat-comp(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
pi1: fst(t),
pi2: snd(t),
type-cat: TypeCat,
op-cat: op-cat(C),
spreadn: spread4,
and: P ∧ Q,
subtype_rel: A ⊆r B,
cat-id: cat-id(C),
compose: f o g,
cand: A c∧ B,
all: ∀x:A. B[x],
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal,
set_wf,
all_wf,
cat-ob_wf,
op-cat_wf,
cat-arrow_wf,
type-cat_wf,
cat-id_wf,
cat-comp_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
dependent_set_memberEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
sqequalRule,
dependent_pairEquality,
lambdaEquality,
setEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
because_Cache,
hypothesis,
functionEquality,
lambdaFormation,
independent_pairFormation,
imageElimination,
extract_by_obid,
isectElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination,
productEquality,
instantiate,
axiomEquality,
isect_memberEquality
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:cat-ob(C)]. \mforall{}[P:U:cat-ob(C) {}\mrightarrow{} (cat-arrow(C) U X) {}\mrightarrow{} \mBbbP{}].
rep-sub-sheaf(C;X;P) \mmember{} Functor(op-cat(C);TypeCat)
supposing \mforall{}A,B:cat-ob(C). \mforall{}g:cat-arrow(C) A B. \mforall{}b:\{b:cat-arrow(C) B X| P B b\} .
(P A (cat-comp(C) A B X g b))
Date html generated:
2017_10_05-AM-00_47_10
Last ObjectModification:
2017_07_28-AM-09_19_34
Theory : small!categories
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