Nuprl Lemma : single-valued-bag-combine
∀[A,B:Type]. ∀[as:bag(A)]. ∀[f:A ⟶ bag(B)].
(single-valued-bag(⋃x∈as.f[x];B)) supposing ((∀a:A. single-valued-bag(f[a];B)) and single-valued-bag(as;A))
Proof
Definitions occuring in Statement :
single-valued-bag: single-valued-bag(b;T),
bag-combine: ⋃x∈bs.f[x],
bag: bag(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
single-valued-bag: single-valued-bag(b;T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
and: P ∧ Q,
squash: ↓T,
exists: ∃x:A. B[x],
prop: ℙ
Lemmas referenced :
bag-member-combine,
and_wf,
equal_wf,
bag_wf,
bag-member_wf,
bag-combine_wf,
all_wf,
single-valued-bag_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
cumulativity,
hypothesis,
productElimination,
independent_isectElimination,
imageElimination,
dependent_functionElimination,
independent_functionElimination,
dependent_set_memberEquality,
independent_pairFormation,
setElimination,
rename,
setEquality,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
axiomEquality,
isect_memberEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[as:bag(A)]. \mforall{}[f:A {}\mrightarrow{} bag(B)].
(single-valued-bag(\mcup{}x\mmember{}as.f[x];B)) supposing
((\mforall{}a:A. single-valued-bag(f[a];B)) and
single-valued-bag(as;A))
Date html generated:
2016_10_25-AM-10_29_28
Last ObjectModification:
2016_07_12-AM-06_45_26
Theory : bags
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