Nuprl Lemma : implies-setmem-piset
∀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 ⊆ Σa:A.B[a])
∧ (∀a:coSet{i:l}. ((a ∈ A) ⇒ (∃b:coSet{i:l}. ((b ∈ B[a]) ∧ ((a,b) ∈ x)))))
∧ (∀a,b1,b2:coSet{i:l}. ((a ∈ A) ⇒ (b1 ∈ B[a]) ⇒ (b2 ∈ B[a]) ⇒ ((a,b1) ∈ x) ⇒ ((a,b2) ∈ x) ⇒ seteq(b1;b2))))
⇒ (x ∈ piset(A;a.B[a])))
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 :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
implies: P ⇒ Q,
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
pi1: fst(t),
pi2: snd(t),
setmem: (x ∈ s),
coWmem: coWmem(a.B[a];z;w),
coW-dom: coW-dom(a.B[a];w),
set-dom: set-dom(s),
mk-coset: mk-coset(T;f),
set-item: set-item(s;x),
cand: A c∧ B,
squash: ↓T,
true: True
Lemmas referenced :
set-item-mem,
set-item_wf,
setmem_wf,
set-dom_wf,
setmem-piset-1,
setsubset_wf,
sigmaset_wf,
orderedpairset_wf,
seteq_wf,
coSet_wf,
subtype_rel_function,
subtype_rel_self,
pi1_wf,
setsubset-iff,
setmem-sigmaset,
subtype_coSet,
coSet_subtype,
setmem-mk-coset,
setmem-coset,
mk-coset_wf,
exists_wf,
subtype_rel-equal,
seteq_functionality,
seteq_weakening,
seteq-orderedpairs-iff,
setmem_functionality_1,
orderedpairset_functionality,
seteq_inversion,
setmem_functionality,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_set_memberEquality_alt,
isectElimination,
universeIsType,
productElimination,
sqequalRule,
lambdaEquality_alt,
applyEquality,
setIsType,
inhabitedIsType,
independent_functionElimination,
productIsType,
functionIsType,
because_Cache,
rename,
functionExtensionality,
instantiate,
cumulativity,
productEquality,
independent_isectElimination,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
independent_pairEquality,
dependent_pairFormation_alt,
independent_pairFormation,
hypothesis_subsumption,
Error :memTop,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
dependent_pairEquality_alt
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 \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)))))
\mwedge{} (\mforall{}a,b1,b2:coSet\{i:l\}.
((a \mmember{} A) {}\mRightarrow{} (b1 \mmember{} B[a]) {}\mRightarrow{} (b2 \mmember{} B[a]) {}\mRightarrow{} ((a,b1) \mmember{} x) {}\mRightarrow{} ((a,b2) \mmember{} x) {}\mRightarrow{} seteq(b1;b2))))
{}\mRightarrow{} (x \mmember{} piset(A;a.B[a])))
Date html generated:
2020_05_20-PM-01_18_58
Last ObjectModification:
2020_01_06-PM-01_24_12
Theory : constructive!set!theory
Home
Index