Nuprl Lemma : setmem-piset-implies
∀A:coSet{i:l}. ∀B:{a:coSet{i:l}| (a ∈ A)} ⟶ coSet{i:l}. ∀x:coSet{i:l}.
((∀a1,a2:coSet{i:l}. ((a1 ∈ A) ⇒ (a2 ∈ A) ⇒ seteq(a1;a2) ⇒ seteq(B[a1];B[a2])))
⇒ (x ∈ piset(A;a.B[a]))
⇒ ((x ⊆ Σa:A.B[a]) ∧ (∀a:coSet{i:l}. ((a ∈ A) ⇒ (∃b:coSet{i:l}. ((b ∈ B[a]) ∧ ((a,b) ∈ x)))))))
Proof
Definitions occuring in Statement :
piset: piset(A;a.B[a]),
sigmaset: Σa:A.B[a],
setsubset: (a ⊆ b),
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],
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},
rev_implies: P ⇐ Q,
cand: A c∧ B,
exists: ∃x:A. B[x],
and: P ∧ Q,
iff: P ⇐⇒ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
co-seteq-iff,
singleset_functionality,
pairset_functionality,
seteq_functionality,
setmem-iff,
singleset_wf,
pairset_wf,
seteq_weakening,
exists_wf,
setmem-sigmaset,
orderedpairset_wf,
setmem_functionality_1,
sigmaset_wf,
setsubset-iff,
seteq_wf,
all_wf,
piset_wf,
coSet_wf,
setmem-piset-1,
set-dom_wf,
setmem_wf,
set-item_wf,
set-item-mem
Rules used in proof :
andLevelFunctionality,
existsFunctionality,
addLevel,
productEquality,
dependent_pairFormation,
functionEquality,
instantiate,
because_Cache,
independent_pairFormation,
independent_functionElimination,
productElimination,
cumulativity,
setEquality,
applyEquality,
lambdaEquality,
sqequalRule,
isectElimination,
dependent_set_memberEquality,
hypothesis,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
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{}x: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{} (x \mmember{} piset(A;a.B[a]))
{}\mRightarrow{} ((x \msubseteq{} \mSigma{}a:A.B[a]) \mwedge{} (\mforall{}a:coSet\{i:l\}. ((a \mmember{} A) {}\mRightarrow{} (\mexists{}b:coSet\{i:l\}. ((b \mmember{} B[a]) \mwedge{} ((a,b) \mmember{} x)))))))
Date html generated:
2018_07_29-AM-10_04_47
Last ObjectModification:
2018_07_18-PM-04_33_48
Theory : constructive!set!theory
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