Nuprl Lemma : bag-mapfilter-wf2
∀[T,A:Type]. ∀[bs:bag(T)]. ∀[p:{b:T| b ↓∈ bs} ⟶ 𝔹]. ∀[f:{x:{b:T| b ↓∈ bs} | ↑p[x]} ⟶ A].
(bag-mapfilter(f;p;bs) ∈ bag(A))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-mapfilter: bag-mapfilter(f;P;bs),
bag: bag(T),
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
all: ∀x:A. B[x],
so_apply: x[s]
Lemmas referenced :
bag-mapfilter_wf,
bag-member_wf,
bag-subtype,
assert_wf,
bool_wf,
bag_wf
Rules used in proof :
cut,
lemma_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
setEquality,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
sqequalRule,
cumulativity,
equalityTransitivity,
equalitySymmetry,
functionEquality,
applyEquality,
because_Cache,
universeEquality,
isect_memberFormation,
introduction,
axiomEquality,
isect_memberEquality
Latex:
\mforall{}[T,A:Type]. \mforall{}[bs:bag(T)]. \mforall{}[p:\{b:T| b \mdownarrow{}\mmember{} bs\} {}\mrightarrow{} \mBbbB{}]. \mforall{}[f:\{x:\{b:T| b \mdownarrow{}\mmember{} bs\} | \muparrow{}p[x]\} {}\mrightarrow{} A].
(bag-mapfilter(f;p;bs) \mmember{} bag(A))
Date html generated:
2016_05_15-PM-02_47_17
Last ObjectModification:
2015_12_27-AM-09_36_15
Theory : bags
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