Nuprl Lemma : co-regext-Regularcoset
The proof that ⌜co-regext(a)⌝ is regular goes through as before, just
using co- versions of everything.⋅
∀a:coSet{i:l}. cRegular(co-regext(a))
Proof
Definitions occuring in Statement :
co-regext: co-regext(a),
Regularcoset: cRegular(A),
coSet: coSet{i:l},
all: ∀x:A. B[x]
Definitions unfolded in proof :
Wsup: Wsup(a;b),
mk-set: f"(T),
ext-eq: A ≡ B,
regextfun: regextfun(f;w),
guard: {T},
exists: ∃x:A. B[x],
top: Top,
co-regext: co-regext(a),
uimplies: b supposing a,
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
mk-coset: mk-coset(T;f),
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
member: t ∈ T,
and: P ∧ Q,
Regularcoset: cRegular(A),
all: ∀x:A. B[x]
Lemmas referenced :
exists_wf,
mk-coset_wf,
seteq_wf,
regextfun_wf,
coW_wf,
subtype_rel_weakening,
set-dom_wf,
coW-ext,
setmem-mk-coset,
coset-relation_wf,
set_wf,
subtype_rel_self,
setmem_wf,
coSet_wf,
subtype_rel_dep_function,
co-regext_wf,
mv-map_wf,
co-regext-lemma,
coSet_subtype,
subtype_coSet,
co-regext-transitive
Rules used in proof :
dependent_pairFormation,
productEquality,
voidEquality,
voidElimination,
isect_memberEquality,
rename,
setElimination,
independent_isectElimination,
setEquality,
universeEquality,
cumulativity,
functionEquality,
lambdaEquality,
because_Cache,
instantiate,
isectElimination,
independent_functionElimination,
productElimination,
sqequalRule,
applyEquality,
hypothesis_subsumption,
hypothesis,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}a:coSet\{i:l\}. cRegular(co-regext(a))
Date html generated:
2018_07_29-AM-10_08_11
Last ObjectModification:
2018_07_21-PM-08_24_03
Theory : constructive!set!theory
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