Nuprl Lemma : setmem-setimages
∀A,B,x:coSet{i:l}.
((x ∈ setimages(A;B))
⇐⇒ ∃f:coSet{i:l}
(function-graph{i:l}(A;_.B;f) ∧ (∀z:coSet{i:l}. ((z ∈ x) ⇐⇒ ∃pr:coSet{i:l}. ((pr ∈ f) ∧ seteq(z;snd(pr)))))))
Proof
Definitions occuring in Statement :
setimages: setimages(A;B),
function-graph: function-graph{i:l}(A;a.B[a];grph),
orderedpair-snd: snd(pr),
setmem: (x ∈ s),
seteq: seteq(s1;s2),
coSet: coSet{i:l},
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
function-graph: function-graph{i:l}(A;a.B[a];grph),
guard: {T},
set-function: set-function{i:l}(s; x.f[x]),
so_apply: x[s],
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
cand: A c∧ B,
member: t ∈ T,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
setimages: setimages(A;B),
all: ∀x:A. B[x]
Lemmas referenced :
snd-orderedpairset,
orderedpair-snd_functionality,
orderedpairset_wf,
setmem-sigmaset,
sigmaset_wf,
setsubset-iff,
setmem_functionality_1,
setmem_functionality,
setmem-imageset,
setunionfun_wf,
co-seteq-iff,
seteq_weakening,
imageset_functionality,
singleset_functionality,
seteq_functionality,
setmem-setunionfun,
singleset_wf,
funset_wf,
setmem-singleset,
setmem-funset,
imageset_wf,
iff_wf,
all_wf,
function-graph_wf,
orderedpair-snd_wf,
seteq_wf,
coSet_wf,
exists_wf,
setmem_wf
Rules used in proof :
rename,
setElimination,
existsLevelFunctionality,
andLevelFunctionality,
independent_functionElimination,
dependent_functionElimination,
existsFunctionality,
impliesFunctionality,
addLevel,
setEquality,
because_Cache,
cumulativity,
productEquality,
lambdaEquality,
sqequalRule,
instantiate,
isectElimination,
extract_by_obid,
introduction,
hypothesis,
hypothesisEquality,
dependent_pairFormation,
thin,
productElimination,
sqequalHypSubstitution,
independent_pairFormation,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}A,B,x:coSet\{i:l\}.
((x \mmember{} setimages(A;B))
\mLeftarrow{}{}\mRightarrow{} \mexists{}f:coSet\{i:l\}
(function-graph\{i:l\}(A;$_{}$.B;f)
\mwedge{} (\mforall{}z:coSet\{i:l\}. ((z \mmember{} x) \mLeftarrow{}{}\mRightarrow{} \mexists{}pr:coSet\{i:l\}. ((pr \mmember{} f) \mwedge{} seteq(z;snd(pr)))))))
Date html generated:
2018_07_29-AM-10_05_55
Last ObjectModification:
2018_07_18-PM-04_43_06
Theory : constructive!set!theory
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