Nuprl Lemma : bag-combine-is-single-if
∀[A,B:Type]. ∀[f:A ⟶ bag(B)]. ∀[bs:bag(A)]. ∀[x:B].
⋃x∈bs.f[x] = {x} ∈ bag(B) supposing ↓∃y:A. ((bs = {y} ∈ bag(A)) ∧ (f[y] = {x} ∈ bag(B)))
Proof
Definitions occuring in Statement :
bag-combine: ⋃x∈bs.f[x],
single-bag: {x},
bag: bag(T),
uimplies: b supposing a,
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,
equal: s = t ∈ T
Definitions unfolded in proof :
squash: ↓T,
exists: ∃x:A. B[x],
and: P ∧ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
all: ∀x:A. B[x],
uimplies: b supposing a
Lemmas referenced :
bag-combine-single-left,
equal_wf,
bag_wf,
bag-combine_wf,
squash_wf,
exists_wf,
single-bag_wf
Rules used in proof :
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
imageElimination,
cut,
productElimination,
thin,
hypothesis,
sqequalRule,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
because_Cache,
lambdaEquality,
applyEquality,
functionExtensionality,
cumulativity,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
equalityTransitivity,
productEquality,
dependent_functionElimination,
functionEquality,
universeEquality,
isect_memberFormation,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[f:A {}\mrightarrow{} bag(B)]. \mforall{}[bs:bag(A)]. \mforall{}[x:B].
\mcup{}x\mmember{}bs.f[x] = \{x\} supposing \mdownarrow{}\mexists{}y:A. ((bs = \{y\}) \mwedge{} (f[y] = \{x\}))
Date html generated:
2017_10_01-AM-08_47_40
Last ObjectModification:
2017_07_26-PM-04_32_05
Theory : bags
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