Nuprl Lemma : bag-member-map3
∀[T,U:Type]. ∀x:U. ∀bs:bag(T). ∀f:{v:T| v ↓∈ bs} ⟶ U. uiff(x ↓∈ bag-map(f;bs);↓∃v:T. (v ↓∈ bs ∧ (x = (f v) ∈ U)))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-map: bag-map(f;bs),
bag: bag(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
squash: ↓T,
exists: ∃x:A. B[x],
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
bag-member: x ↓∈ bs
Lemmas referenced :
bag-member_wf,
bag-map-member-wf,
bag-member-map,
bag-subtype,
equal_wf,
squash_wf,
exists_wf,
bag-subtype2,
iff_weakening_uiff,
bag_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
hypothesis,
setEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
imageElimination,
productElimination,
setElimination,
rename,
dependent_pairFormation,
productEquality,
applyEquality,
dependent_set_memberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
universeIsType,
lambdaEquality,
independent_functionElimination,
independent_isectElimination,
functionIsType,
setIsType,
inhabitedIsType,
universeEquality
Latex:
\mforall{}[T,U:Type].
\mforall{}x:U. \mforall{}bs:bag(T). \mforall{}f:\{v:T| v \mdownarrow{}\mmember{} bs\} {}\mrightarrow{} U. uiff(x \mdownarrow{}\mmember{} bag-map(f;bs);\mdownarrow{}\mexists{}v:T. (v \mdownarrow{}\mmember{} bs \mwedge{} (x = (f v)))\000C)
Date html generated:
2019_10_15-AM-11_02_23
Last ObjectModification:
2018_09_27-AM-11_19_30
Theory : bags
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