Nuprl Lemma : setmem-unionfun-implies
∀s:coSet{i:l}. ∀f:{x:coSet{i:l}| (x ∈ s)} ⟶ coSet{i:l}. ∀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],
setmem: (x ∈ s),
coSet: coSet{i:l},
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
cand: A c∧ B,
and: P ∧ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
exists: ∃x:A. B[x],
top: Top,
uall: ∀[x:A]. B[x],
mk-coset: mk-coset(T;f),
setunionfun: ⋃x∈s.f[x],
subtype_rel: A ⊆r B,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
set-item-mem,
set-item_wf,
setmem_functionality_1,
mk-coset_wf,
setmem-coset,
coSet_wf,
setunionfun_wf,
setmem_wf,
setmem-mk-coset,
coSet_subtype,
subtype_coSet
Rules used in proof :
independent_functionElimination,
dependent_set_memberEquality,
universeEquality,
because_Cache,
functionExtensionality,
productEquality,
independent_pairFormation,
dependent_functionElimination,
dependent_pairFormation,
functionEquality,
cumulativity,
setEquality,
lambdaEquality,
voidEquality,
voidElimination,
isect_memberEquality,
isectElimination,
rename,
thin,
productElimination,
sqequalRule,
sqequalHypSubstitution,
applyEquality,
hypothesisEquality,
hypothesis,
extract_by_obid,
introduction,
cut,
hypothesis_subsumption,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}s:coSet\{i:l\}. \mforall{}f:\{x:coSet\{i:l\}| (x \mmember{} s)\} {}\mrightarrow{} coSet\{i:l\}. \mforall{}y:coSet\{i:l\}.
((y \mmember{} \mcup{}x\mmember{}s.f[x]) {}\mRightarrow{} (\mexists{}x:coSet\{i:l\}. ((x \mmember{} s) \mwedge{} (y \mmember{} f[x]))))
Date html generated:
2018_07_29-AM-09_52_51
Last ObjectModification:
2018_07_18-PM-02_36_21
Theory : constructive!set!theory
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