Nuprl Lemma : setmem-imageset
∀B,f,x:coSet{i:l}. ((x ∈ imageset(B;f)) ⇐⇒ (x ∈ B) ∧ (∃pr:coSet{i:l}. ((pr ∈ f) ∧ seteq(x;snd(pr)))))
Proof
Definitions occuring in Statement :
imageset: imageset(B;f),
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 :
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q,
existssetmem: ∃a∈A.P[a],
set-predicate: set-predicate{i:l}(s;a.P[a]),
implies: P ⇒ Q,
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
iff: P ⇐⇒ Q,
imageset: imageset(B;f),
all: ∀x:A. B[x]
Lemmas referenced :
orderedpair-snd_functionality,
existssetmem-iff,
exists_wf,
iff_wf,
sub-set_wf,
seteq_weakening,
set-item_wf,
seteq_functionality,
setmem_wf,
coSet_wf,
orderedpair-snd_wf,
seteq_wf,
existssetmem_wf,
setmem-sub-coset
Rules used in proof :
andLevelFunctionality,
productEquality,
instantiate,
existsLevelFunctionality,
promote_hyp,
levelHypothesis,
because_Cache,
existsFunctionality,
independent_functionElimination,
cumulativity,
setEquality,
hypothesis,
rename,
setElimination,
lambdaEquality,
sqequalRule,
isectElimination,
hypothesisEquality,
dependent_functionElimination,
extract_by_obid,
introduction,
impliesFunctionality,
independent_pairFormation,
thin,
productElimination,
sqequalHypSubstitution,
addLevel,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}B,f,x:coSet\{i:l\}.
((x \mmember{} imageset(B;f)) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} B) \mwedge{} (\mexists{}pr:coSet\{i:l\}. ((pr \mmember{} f) \mwedge{} seteq(x;snd(pr)))))
Date html generated:
2018_07_29-AM-10_05_40
Last ObjectModification:
2018_07_18-PM-03_19_38
Theory : constructive!set!theory
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