Nuprl Lemma : presheaf-fun-family-comp
∀C:SmallCategory. ∀X,Delta:ps_context{j:l}(C). ∀s:psc_map{j:l}(C; Delta; X). ∀I,J:cat-ob(C). ∀f:cat-arrow(C) J I.
∀a:Delta(I). ∀A,B:{X ⊢ _}. ∀w:presheaf-fun-family(C; X; A; B; I; (s)a).
(λK,g. (w K (cat-comp(C) K J I g f)) ∈ presheaf-fun-family(C; X; A; B; J; (s)f(a)))
Proof
Definitions occuring in Statement :
presheaf-fun-family: presheaf-fun-family(C; X; A; B; I; a),
presheaf-type: {X ⊢ _},
pscm-ap: (s)x,
psc_map: A ⟶ B,
psc-restriction: f(s),
I_set: A(I),
ps_context: __⊢,
all: ∀x:A. B[x],
member: t ∈ T,
apply: f a,
lambda: λx.A[x],
cat-comp: cat-comp(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
presheaf-fun-family: presheaf-fun-family(C; X; A; B; I; a),
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
true: True,
prop: ℙ
Lemmas referenced :
cat-comp_wf,
subtype_rel_dep_function,
presheaf-type-at_wf,
psc-restriction_wf,
pscm-ap_wf,
subtype_rel-equal,
equal_wf,
pscm-ap-restriction,
iff_weakening_equal,
psc-restriction-comp,
cat-arrow_wf,
presheaf-type-ap-morph_wf,
squash_wf,
true_wf,
I_set_wf,
subtype_rel_self,
presheaf-fun-family_wf,
presheaf-type_wf,
cat-ob_wf,
psc_map_wf,
small-category-cumulativity-2,
ps_context_wf,
small-category_wf,
istype-universe,
cat-comp-assoc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality_alt,
lambdaEquality_alt,
applyEquality,
hypothesisEquality,
introduction,
extract_by_obid,
isectElimination,
hypothesis,
because_Cache,
sqequalRule,
universeIsType,
independent_isectElimination,
imageElimination,
instantiate,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_functionElimination,
natural_numberEquality,
functionIsType,
equalityIstype,
inhabitedIsType,
universeEquality,
hyp_replacement
Latex:
\mforall{}C:SmallCategory. \mforall{}X,Delta:ps\_context\{j:l\}(C). \mforall{}s:psc\_map\{j:l\}(C; Delta; X). \mforall{}I,J:cat-ob(C).
\mforall{}f:cat-arrow(C) J I. \mforall{}a:Delta(I). \mforall{}A,B:\{X \mvdash{} \_\}. \mforall{}w:presheaf-fun-family(C; X; A; B; I; (s)a).
(\mlambda{}K,g. (w K (cat-comp(C) K J I g f)) \mmember{} presheaf-fun-family(C; X; A; B; J; (s)f(a)))
Date html generated:
2020_05_20-PM-01_29_03
Last ObjectModification:
2020_04_02-PM-03_01_20
Theory : presheaf!models!of!type!theory
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