Nuprl Lemma : regext-Regularset
∀a:Set{i:l}. Regular(regext(a))
Proof
Definitions occuring in Statement :
regext: regext(a),
Regularset: Regular(A),
Set: Set{i:l},
all: ∀x:A. B[x]
Definitions unfolded in proof :
set-relation: SetRelation(R),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
so_apply: x[s],
so_lambda: λ2x.t[x],
onto-map: R:(A ─>> B),
mv-map: R:(A ⇒ B),
cand: A c∧ B,
coset-relation: coSetRelation(R),
exists: ∃x:A. B[x],
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
implies: P ⇒ Q,
Regularset: Regular(A),
and: P ∧ Q,
Regularcoset: cRegular(A),
subtype_rel: A ⊆r B,
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
seteq_inversion,
setmem_functionality_1,
set-relation_wf,
onto-map_wf,
set_wf,
subtype_rel_self,
coSet-subtype-Set,
subtype_rel_dep_function,
mv-map_wf,
seteq_wf,
coSet_wf,
regext_wf2,
coSet-mem-Set-implies-Set,
Set_wf,
regext_wf,
setmem_wf,
set-subtype-coSet,
regext-Regularcoset
Rules used in proof :
setEquality,
functionEquality,
instantiate,
dependent_pairFormation,
independent_isectElimination,
functionExtensionality,
universeEquality,
cumulativity,
because_Cache,
isectElimination,
productEquality,
lambdaEquality,
independent_functionElimination,
independent_pairFormation,
productElimination,
sqequalRule,
applyEquality,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}a:Set\{i:l\}. Regular(regext(a))
Date html generated:
2018_07_29-AM-10_07_41
Last ObjectModification:
2018_07_20-PM-01_48_22
Theory : constructive!set!theory
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