Nuprl Lemma : member-f-union
∀[T,A:Type]. ∀[eqt:EqDecider(T)]. ∀[eqa:EqDecider(A)]. ∀[g:T ⟶ fset(A)]. ∀[s:fset(T)]. ∀[a:A].
uiff(a ∈ f-union(eqt;eqa;s;x.g[x]);↓∃x:T. (x ∈ s ∧ a ∈ g[x]))
Proof
Definitions occuring in Statement :
f-union: f-union(domeq;rngeq;s;x.g[x]),
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
so_apply: x[s],
exists: ∃x:A. B[x],
squash: ↓T,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
fset: fset(T),
exists: ∃x:A. B[x],
quotient: x,y:A//B[x; y],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
guard: {T},
fset-member: a ∈ s,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
true: True,
false: False,
not: ¬A
Lemmas referenced :
fset-member_wf,
f-union_wf,
fset-member_witness,
squash_wf,
exists_wf,
fset_wf,
deq_wf,
equal-wf-base,
list_wf,
set-equal_wf,
member-f-union-aux,
l_exists_iff,
l_member_wf,
list_subtype_fset,
assert-deq-member,
decidable__fset-member,
subtype_base_sq,
int_subtype_base,
set-equal-reflex,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
sqequalHypSubstitution,
imageElimination,
sqequalRule,
imageMemberEquality,
hypothesisEquality,
thin,
baseClosed,
extract_by_obid,
isectElimination,
cumulativity,
lambdaEquality,
applyEquality,
functionExtensionality,
independent_functionElimination,
productEquality,
productElimination,
independent_pairEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
pointwiseFunctionalityForEquality,
pertypeElimination,
dependent_functionElimination,
setElimination,
rename,
setEquality,
dependent_pairFormation,
independent_isectElimination,
unionElimination,
instantiate,
intEquality,
natural_numberEquality,
voidElimination,
promote_hyp,
lambdaFormation,
pointwiseFunctionality
Latex:
\mforall{}[T,A:Type]. \mforall{}[eqt:EqDecider(T)]. \mforall{}[eqa:EqDecider(A)]. \mforall{}[g:T {}\mrightarrow{} fset(A)]. \mforall{}[s:fset(T)]. \mforall{}[a:A].
uiff(a \mmember{} f-union(eqt;eqa;s;x.g[x]);\mdownarrow{}\mexists{}x:T. (x \mmember{} s \mwedge{} a \mmember{} g[x]))
Date html generated:
2017_04_17-AM-09_19_10
Last ObjectModification:
2017_02_27-PM-05_22_45
Theory : finite!sets
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