Nuprl Lemma : regext-lemma
∀T:Type. ∀f:T ⟶ Set{i:l}. ∀B:Set{i:l}.
((∃t:T. ∃g:set-dom(f t) ⟶ Set{i:l}. seteq(B;g"(set-dom(f t))))
⇒ (∀R:Set{i:l} ⟶ Set{i:l} ⟶ ℙ'
(SetRelation(R)
⇒ R:(B ⇒ regext(f"(T)))
⇒ (∃b:Set{i:l}. ((b ∈ regext(f"(T))) ∧ R:(B ⇒ b) ∧ R:(B ─>> b))))))
Proof
Definitions occuring in Statement :
regext: regext(a),
onto-map: R:(A ─>> B),
mv-map: R:(A ⇒ B),
set-relation: SetRelation(R),
mk-set: f"(T),
Set: Set{i:l},
setmem: (x ∈ s),
seteq: seteq(s1;s2),
set-dom: set-dom(s),
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
onto-map: R:(A ─>> B),
mv-map: R:(A ⇒ B),
set-relation: SetRelation(R),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
cand: A c∧ B,
coset-relation: coSetRelation(R),
and: P ∧ Q,
Wsup: Wsup(a;b),
mk-set: f"(T),
mk-coset: mk-coset(T;f),
exists: ∃x:A. B[x],
uimplies: b supposing a,
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
onto-map_wf,
seteq_inversion,
setmem_functionality_1,
regext_wf2,
coSet-mem-Set-implies-Set,
mk-coset_wf,
seteq_wf,
set-dom_wf,
exists_wf,
set-relation_wf,
set_wf,
subtype_rel_self,
coSet-subtype-Set,
setmem_wf,
coSet_wf,
Set_wf,
subtype_rel_dep_function,
mk-set_wf,
regext_wf,
mv-map_wf,
set-subtype-coSet,
regext-lemma1
Rules used in proof :
independent_pairFormation,
productEquality,
dependent_pairFormation,
productElimination,
independent_isectElimination,
setEquality,
universeEquality,
functionEquality,
lambdaEquality,
instantiate,
isectElimination,
independent_functionElimination,
cumulativity,
sqequalRule,
because_Cache,
hypothesis,
applyEquality,
functionExtensionality,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}T:Type. \mforall{}f:T {}\mrightarrow{} Set\{i:l\}. \mforall{}B:Set\{i:l\}.
((\mexists{}t:T. \mexists{}g:set-dom(f t) {}\mrightarrow{} Set\{i:l\}. seteq(B;g"(set-dom(f t))))
{}\mRightarrow{} (\mforall{}R:Set\{i:l\} {}\mrightarrow{} Set\{i:l\} {}\mrightarrow{} \mBbbP{}'
(SetRelation(R)
{}\mRightarrow{} R:(B {}\mRightarrow{} regext(f"(T)))
{}\mRightarrow{} (\mexists{}b:Set\{i:l\}. ((b \mmember{} regext(f"(T))) \mwedge{} R:(B {}\mRightarrow{} b) \mwedge{} R:(B {}>> b))))))
Date html generated:
2018_07_29-AM-10_07_34
Last ObjectModification:
2018_07_20-PM-05_47_50
Theory : constructive!set!theory
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