Nuprl Lemma : bmsexists_char
∀s:DSet. ∀f:|s| ⟶ 𝔹. ∀a:MSet{s}. ((∃x:|s|. ((↑(x ∈b a)) ∧ (↑f[x]))) ⇒ (↑(∃b{s} x ∈ a. f[x])))
Proof
Definitions occuring in Statement :
mset_for: mset_for,
mset_mem: mset_mem,
mset: MSet{s},
assert: ↑b,
bool: 𝔹,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
bor_mon: <𝔹,∨b>,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
dset: DSet,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
sq_stable: SqStable(P),
mset: MSet{s},
bor_mon: <𝔹,∨b>,
grp_car: |g|,
pi1: fst(t),
quotient: x,y:A//B[x; y],
squash: ↓T,
mset_for: mset_for,
mset_mem: mset_mem,
bexists: bexists,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
sq_stable__all,
exists_wf,
set_car_wf,
assert_wf,
mset_mem_wf,
mset_for_wf,
bor_mon_wf,
sq_stable_from_decidable,
decidable__assert,
squash_wf,
abmonoid_subtype_iabmonoid,
bool_wf,
list_wf,
permr_wf,
equal_wf,
equal-wf-base,
mset_wf,
dset_wf,
bexists_char,
mem_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
productEquality,
dependent_functionElimination,
applyEquality,
functionExtensionality,
because_Cache,
independent_functionElimination,
pointwiseFunctionalityForEquality,
functionEquality,
pertypeElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed
Latex:
\mforall{}s:DSet. \mforall{}f:|s| {}\mrightarrow{} \mBbbB{}. \mforall{}a:MSet\{s\}. ((\mexists{}x:|s|. ((\muparrow{}(x \mmember{}\msubb{} a)) \mwedge{} (\muparrow{}f[x]))) {}\mRightarrow{} (\muparrow{}(\mexists{}\msubb{}\{s\} x \mmember{} a. f[x])))
Date html generated:
2017_10_01-AM-09_59_24
Last ObjectModification:
2017_03_03-PM-01_00_34
Theory : mset
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