Nuprl Lemma : setmem-closure-set
∀B:Set{i:l}. ∀Y:Set{i:l} ⟶ Set{i:l}. ∀x:Set{i:l}.
((∀b:Set{i:l}. ((b ∈ B) ⇒ set-function{i:l}(setimages(b;x); A.Y A)))
⇒ (∀z:Set{i:l}
((z ∈ closure-set(B;Y;x)) ⇐⇒ ∃b:coSet{i:l}. ((b ∈ B) ∧ (∃A:coSet{i:l}. ((A ∈ setimages(b;x)) ∧ (z ∈ Y A)))))))
Proof
Definitions occuring in Statement :
closure-set: closure-set(B;Y;x),
setimages: setimages(A;B),
set-function: set-function{i:l}(s; x.f[x]),
Set: Set{i:l},
setmem: (x ∈ s),
coSet: coSet{i:l},
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
guard: {T},
rev_implies: P ⇐ Q,
set-function: set-function{i:l}(s; x.f[x]),
so_apply: x[s],
exists: ∃x:A. B[x],
uimplies: b supposing a,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
member: t ∈ T,
and: P ∧ Q,
iff: P ⇐⇒ Q,
closure-set: closure-set(B;Y;x),
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
set-function_wf,
all_wf,
seteq_wf,
iff_wf,
setimages_functionality,
seteq_weakening,
setmem_functionality,
exists_wf,
Set_wf,
setunionfun_wf2,
seteq-iff,
setimages_wf2,
coSet-mem-Set-implies-Set,
setimages_wf,
setunionfun_wf,
coSet_wf,
setmem_wf,
set-subtype-coSet,
setmem-setunionfun
Rules used in proof :
functionEquality,
existsLevelFunctionality,
andLevelFunctionality,
existsFunctionality,
productEquality,
instantiate,
independent_functionElimination,
universeEquality,
setEquality,
dependent_pairFormation,
independent_isectElimination,
functionExtensionality,
rename,
setElimination,
because_Cache,
isectElimination,
cumulativity,
lambdaEquality,
sqequalRule,
hypothesis,
applyEquality,
hypothesisEquality,
dependent_functionElimination,
extract_by_obid,
introduction,
impliesFunctionality,
independent_pairFormation,
thin,
productElimination,
sqequalHypSubstitution,
addLevel,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}B:Set\{i:l\}. \mforall{}Y:Set\{i:l\} {}\mrightarrow{} Set\{i:l\}. \mforall{}x:Set\{i:l\}.
((\mforall{}b:Set\{i:l\}. ((b \mmember{} B) {}\mRightarrow{} set-function\{i:l\}(setimages(b;x); A.Y A)))
{}\mRightarrow{} (\mforall{}z:Set\{i:l\}
((z \mmember{} closure-set(B;Y;x))
\mLeftarrow{}{}\mRightarrow{} \mexists{}b:coSet\{i:l\}. ((b \mmember{} B) \mwedge{} (\mexists{}A:coSet\{i:l\}. ((A \mmember{} setimages(b;x)) \mwedge{} (z \mmember{} Y A)))))))
Date html generated:
2018_07_29-AM-10_10_06
Last ObjectModification:
2018_07_18-PM-09_08_33
Theory : constructive!set!theory
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