Nuprl Lemma : poset-cat-arrow_subtype
∀[I,J:Cname List].
∀[x,y:cat-ob(poset-cat(I))]. ((cat-arrow(poset-cat(I)) x y) ⊆r (cat-arrow(poset-cat(J)) x y))
supposing nameset(J) ⊆r nameset(I)
Proof
Definitions occuring in Statement :
poset-cat: poset-cat(J),
nameset: nameset(L),
coordinate_name: Cname,
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
list: T List,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
apply: f a
Definitions unfolded in proof :
poset-cat: poset-cat(J),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
pi1: fst(t),
pi2: snd(t),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
name-morph: name-morph(I;J),
prop: ℙ,
so_apply: x[s]
Lemmas referenced :
subtype_rel_dep_function,
nameset_wf,
assert_wf,
le_int_wf,
subtype_rel_self,
all_wf,
name-morph_wf,
nil_wf,
coordinate_name_wf,
subtype_rel_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
lambdaEquality,
sqequalHypSubstitution,
hypothesisEquality,
applyEquality,
lemma_by_obid,
isectElimination,
thin,
hypothesis,
setElimination,
rename,
because_Cache,
independent_isectElimination,
lambdaFormation,
axiomEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[I,J:Cname List].
\mforall{}[x,y:cat-ob(poset-cat(I))]. ((cat-arrow(poset-cat(I)) x y) \msubseteq{}r (cat-arrow(poset-cat(J)) x y))
supposing nameset(J) \msubseteq{}r nameset(I)
Date html generated:
2016_06_16-PM-06_52_17
Last ObjectModification:
2015_12_28-PM-04_23_00
Theory : cubical!sets
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