Nuprl Lemma : setmem-sigmaset
∀A:coSet{i:l}. ∀B:{a:coSet{i:l}| (a ∈ A)} ⟶ coSet{i:l}.
((∀a1,a2:coSet{i:l}. ((a1 ∈ A) ⇒ (a2 ∈ A) ⇒ seteq(a1;a2) ⇒ seteq(B[a1];B[a2])))
⇒ (∀x:coSet{i:l}. ((x ∈ Σa:A.B[a]) ⇐⇒ ∃a:coSet{i:l}. ((a ∈ A) ∧ (∃b:coSet{i:l}. ((b ∈ B[a]) ∧ seteq(x;(a,b))))))))
Proof
Definitions occuring in Statement :
sigmaset: Σa:A.B[a],
orderedpairset: (a,b),
setmem: (x ∈ s),
seteq: seteq(s1;s2),
coSet: coSet{i:l},
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
orderedpairset: (a,b),
guard: {T},
cand: A c∧ B,
top: Top,
mk-coset: mk-coset(T;f),
sigmaset: Σa:A.B[a],
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
singleset_functionality,
pairset_functionality,
singleset_wf,
pairset_wf,
seteq_functionality,
set-dom_wf,
seteq_weakening,
setmem_functionality,
setmem-iff,
setmem_functionality_1,
set-item-mem,
mk-coset_wf,
set-item_wf,
setmem-coset,
setmem-mk-coset,
coSet_subtype,
subtype_coSet,
all_wf,
orderedpairset_wf,
seteq_wf,
exists_wf,
coSet_wf,
sigmaset_wf,
setmem_wf
Rules used in proof :
spreadEquality,
dependent_pairEquality,
universeEquality,
functionExtensionality,
independent_functionElimination,
rename,
dependent_functionElimination,
dependent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
hypothesis_subsumption,
because_Cache,
functionEquality,
dependent_set_memberEquality,
productEquality,
instantiate,
productElimination,
cumulativity,
hypothesis,
setEquality,
applyEquality,
lambdaEquality,
sqequalRule,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}A:coSet\{i:l\}. \mforall{}B:\{a:coSet\{i:l\}| (a \mmember{} A)\} {}\mrightarrow{} coSet\{i:l\}.
((\mforall{}a1,a2:coSet\{i:l\}. ((a1 \mmember{} A) {}\mRightarrow{} (a2 \mmember{} A) {}\mRightarrow{} seteq(a1;a2) {}\mRightarrow{} seteq(B[a1];B[a2])))
{}\mRightarrow{} (\mforall{}x:coSet\{i:l\}
((x \mmember{} \mSigma{}a:A.B[a])
\mLeftarrow{}{}\mRightarrow{} \mexists{}a:coSet\{i:l\}. ((a \mmember{} A) \mwedge{} (\mexists{}b:coSet\{i:l\}. ((b \mmember{} B[a]) \mwedge{} seteq(x;(a,b))))))))
Date html generated:
2018_07_29-AM-10_04_16
Last ObjectModification:
2018_07_18-PM-03_53_40
Theory : constructive!set!theory
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