Nuprl Lemma : regext-lemma1
∀T:Type. ∀f:T ⟶ coSet{i:l}. ∀B:coSet{i:l}.
((∃t:T. ∃g:set-dom(f t) ⟶ coSet{i:l}. seteq(B;mk-coset(set-dom(f t);g)))
⇒ (∀R:coSet{i:l} ⟶ coSet{i:l} ⟶ ℙ'
(coSetRelation(R)
⇒ R:(B ⇒ regext(mk-coset(T;f)))
⇒ (∃b:coSet{i:l}. ((b ∈ regext(mk-coset(T;f))) ∧ R:(B ⇒ b) ∧ R:(B ─>> b))))))
Proof
Definitions occuring in Statement :
regext: regext(a),
onto-map: R:(A ─>> B),
mv-map: R:(A ⇒ B),
coset-relation: coSetRelation(R),
setmem: (x ∈ s),
seteq: seteq(s1;s2),
set-dom: set-dom(s),
mk-coset: mk-coset(T;f),
coSet: coSet{i:l},
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),
regextfun: regextfun(f;w),
cand: A c∧ B,
mk-set: f"(T),
pi1: fst(t),
rev_implies: P ⇐ Q,
guard: {T},
Wsup: Wsup(a;b),
top: Top,
uimplies: b supposing a,
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
mk-coset: mk-coset(T;f),
regext: regext(a),
exists: ∃x:A. B[x],
mv-map: R:(A ⇒ B),
coset-relation: coSetRelation(R),
prop: ℙ,
and: P ∧ Q,
iff: P ⇐⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
seteq_inversion,
setmem-mk-coset,
onto-map_wf,
equal_wf,
setmem_functionality,
all_wf,
W_wf,
setmem-coset,
Wsup_wf,
W-subtype-coW,
regextfun_wf,
seteq_weakening,
setmem-mk-set-sq,
seteq_wf,
set-dom_wf,
exists_wf,
coset-relation_wf,
set_wf,
subtype_rel_self,
coSet_wf,
subtype_rel_dep_function,
mv-map_wf,
setmem_wf,
setsubset-iff,
regext_wf,
transitive-set-iff,
mk-coset_wf,
regext-transitive
Rules used in proof :
productEquality,
independent_pairFormation,
functionExtensionality,
equalitySymmetry,
equalityTransitivity,
dependent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
rename,
setElimination,
independent_isectElimination,
setEquality,
universeEquality,
cumulativity,
functionEquality,
lambdaEquality,
because_Cache,
instantiate,
applyEquality,
sqequalRule,
promote_hyp,
allFunctionality,
independent_functionElimination,
productElimination,
hypothesis,
hypothesisEquality,
isectElimination,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}T:Type. \mforall{}f:T {}\mrightarrow{} coSet\{i:l\}. \mforall{}B:coSet\{i:l\}.
((\mexists{}t:T. \mexists{}g:set-dom(f t) {}\mrightarrow{} coSet\{i:l\}. seteq(B;mk-coset(set-dom(f t);g)))
{}\mRightarrow{} (\mforall{}R:coSet\{i:l\} {}\mrightarrow{} coSet\{i:l\} {}\mrightarrow{} \mBbbP{}'
(coSetRelation(R)
{}\mRightarrow{} R:(B {}\mRightarrow{} regext(mk-coset(T;f)))
{}\mRightarrow{} (\mexists{}b:coSet\{i:l\}. ((b \mmember{} regext(mk-coset(T;f))) \mwedge{} R:(B {}\mRightarrow{} b) \mwedge{} R:(B {}>> b))))))
Date html generated:
2018_07_29-AM-10_07_29
Last ObjectModification:
2018_07_20-PM-05_48_17
Theory : constructive!set!theory
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