Nuprl Lemma : setmem-setunionfun
∀s:coSet{i:l}. ∀f:{x:coSet{i:l}| (x ∈ s)} ⟶ coSet{i:l}.
(set-function{i:l}(s; x.f[x]) ⇒ (∀y:coSet{i:l}. ((y ∈ ⋃x∈s.f[x]) ⇐⇒ ∃x:coSet{i:l}. ((x ∈ s) ∧ (y ∈ f[x])))))
Proof
Definitions occuring in Statement :
setunionfun: ⋃x∈s.f[x],
set-function: set-function{i:l}(s; x.f[x]),
setmem: (x ∈ s),
coSet: coSet{i:l},
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
setunionfun: ⋃x∈s.f[x],
guard: {T},
pi2: snd(t),
pi1: fst(t),
set-dom: set-dom(s),
set-item: set-item(s;x),
set-function: set-function{i:l}(s; x.f[x]),
top: Top,
exists: ∃x:A. B[x],
mk-coset: mk-coset(T;f),
rev_implies: P ⇐ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
member: t ∈ T,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
set-item_wf,
set-dom_wf,
seteq_inversion,
seteq_weakening,
setmem_functionality,
seteq_wf,
setmem-iff,
setmem-coset,
setmem_functionality_1,
setmem-mk-coset,
mk-coset_wf,
setmem-unionfun-implies,
set-function_wf,
exists_wf,
coSet_wf,
setunionfun_wf,
setmem_wf,
coSet_subtype,
subtype_coSet
Rules used in proof :
dependent_pairEquality,
dependent_pairFormation,
universeEquality,
functionExtensionality,
because_Cache,
voidEquality,
voidElimination,
isect_memberEquality,
independent_functionElimination,
dependent_functionElimination,
functionEquality,
dependent_set_memberEquality,
productEquality,
instantiate,
cumulativity,
setEquality,
lambdaEquality,
isectElimination,
rename,
thin,
productElimination,
sqequalRule,
sqequalHypSubstitution,
applyEquality,
hypothesisEquality,
hypothesis,
extract_by_obid,
introduction,
cut,
hypothesis_subsumption,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}s:coSet\{i:l\}. \mforall{}f:\{x:coSet\{i:l\}| (x \mmember{} s)\} {}\mrightarrow{} coSet\{i:l\}.
(set-function\{i:l\}(s; x.f[x])
{}\mRightarrow{} (\mforall{}y:coSet\{i:l\}. ((y \mmember{} \mcup{}x\mmember{}s.f[x]) \mLeftarrow{}{}\mRightarrow{} \mexists{}x:coSet\{i:l\}. ((x \mmember{} s) \mwedge{} (y \mmember{} f[x])))))
Date html generated:
2018_07_29-AM-09_52_54
Last ObjectModification:
2018_07_18-PM-02_44_57
Theory : constructive!set!theory
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