Nuprl Lemma : setmem-piset-1
∀A:coSet{i:l}. ∀B:{a:coSet{i:l}| (a ∈ A)} ⟶ coSet{i:l}. ∀x:coSet{i:l}.
((x ∈ piset(A;a.B[a]))
⇐⇒ ∃f:t:set-dom(A) ⟶ set-dom(B[set-item(A;t)])
∀z:coSet{i:l}. ((z ∈ x) ⇐⇒ ∃t:set-dom(A). seteq(z;(set-item(A;t),set-item(B[set-item(A;t)];f t)))))
Proof
Definitions occuring in Statement :
piset: piset(A;a.B[a]),
orderedpairset: (a,b),
setmem: (x ∈ s),
seteq: seteq(s1;s2),
set-item: set-item(s;x),
set-dom: set-dom(s),
coSet: coSet{i:l},
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uimplies: b supposing a,
pi2: snd(t),
pi1: fst(t),
set-dom: set-dom(s),
set-item: set-item(s;x),
guard: {T},
top: Top,
mk-coset: mk-coset(T;f),
piset: piset(A;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,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
subtype_rel-equal,
co-seteq-iff,
subtype_rel_self,
seteq_weakening,
setmem_functionality,
mk-coset_wf,
setmem-coset,
setmem-mk-coset,
coSet_subtype,
subtype_coSet,
orderedpairset_wf,
seteq_wf,
iff_wf,
all_wf,
set-item_wf,
set-item-mem,
set-dom_wf,
exists_wf,
coSet_wf,
piset_wf,
setmem_wf
Rules used in proof :
setElimination,
independent_isectElimination,
impliesLevelFunctionality,
andLevelFunctionality,
allLevelFunctionality,
independent_functionElimination,
impliesFunctionality,
allFunctionality,
existsFunctionality,
addLevel,
dependent_pairFormation,
rename,
voidEquality,
voidElimination,
isect_memberEquality,
hypothesis_subsumption,
dependent_set_memberEquality,
dependent_functionElimination,
because_Cache,
functionExtensionality,
universeEquality,
functionEquality,
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{}x:coSet\{i:l\}.
((x \mmember{} piset(A;a.B[a]))
\mLeftarrow{}{}\mRightarrow{} \mexists{}f:t:set-dom(A) {}\mrightarrow{} set-dom(B[set-item(A;t)])
\mforall{}z:coSet\{i:l\}
((z \mmember{} x) \mLeftarrow{}{}\mRightarrow{} \mexists{}t:set-dom(A). seteq(z;(set-item(A;t),set-item(B[set-item(A;t)];f t)))))
Date html generated:
2018_07_29-AM-10_04_31
Last ObjectModification:
2018_07_18-PM-04_20_32
Theory : constructive!set!theory
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