Nuprl Lemma : mset_mem_map
∀s,s':DSet. ∀f:|s| ⟶ |s'|. ∀x:|s|. ∀a:MSet{s}. ((↑(x ∈b a)) ⇒ (↑((f x) ∈b msmap{s,s'}(f;a))))
Proof
Definitions occuring in Statement :
mset_map: msmap{s,s'}(f;a),
mset_mem: mset_mem,
mset: MSet{s},
assert: ↑b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
mset_mem: mset_mem,
mk_mset: mk_mset(as),
uall: ∀[x:A]. B[x],
member: t ∈ T,
dset: DSet,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
so_apply: x[s],
mset_map: msmap{s,s'}(f;a),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
mem_map,
list_wf,
all_wf,
decidable__assert,
sq_stable_from_decidable,
sq_stable__all,
assert_wf,
all_mset_elim,
mset_map_char,
dset_wf,
mset_wf,
true_wf,
squash_wf,
set_car_wf,
map_wf,
mem_wf,
mk_mset_wf,
mset_map_wf,
mset_mem_wf,
assert_functionality_wrt_uiff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
addLevel,
allFunctionality,
impliesFunctionality,
sqequalHypSubstitution,
sqequalRule,
hypothesis,
lemma_by_obid,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
setElimination,
rename,
independent_isectElimination,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
functionEquality,
independent_functionElimination,
levelHypothesis,
allLevelFunctionality,
impliesLevelFunctionality
Latex:
\mforall{}s,s':DSet. \mforall{}f:|s| {}\mrightarrow{} |s'|. \mforall{}x:|s|. \mforall{}a:MSet\{s\}. ((\muparrow{}(x \mmember{}\msubb{} a)) {}\mRightarrow{} (\muparrow{}((f x) \mmember{}\msubb{} msmap\{s,s'\}(f;a))))
Date html generated:
2016_05_16-AM-07_50_09
Last ObjectModification:
2016_01_16-PM-11_39_22
Theory : mset
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