Nuprl Lemma : member-fset-constrained-image-iff
∀[T,A:Type]. ∀[eqt:EqDecider(T)]. ∀[eqa:EqDecider(A)]. ∀[f:T ⟶ A]. ∀[P:A ⟶ 𝔹]. ∀[s:fset(T)]. ∀[a:A].
uiff(a ∈ f"(s) s.t. P;↓∃x:T. (x ∈ s ∧ (a = (f x) ∈ A) ∧ (↑(P a))))
Proof
Definitions occuring in Statement :
fset-constrained-image: f"(s) s.t. P,
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
assert: ↑b,
bool: 𝔹,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
fset-constrained-image: f"(s) s.t. P,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
squash: ↓T,
exists: ∃x:A. B[x],
prop: ℙ,
cand: A c∧ B,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
top: Top,
not: ¬A,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
fset-member_wf,
equal_wf,
assert_wf,
squash_wf,
exists_wf,
ifthenelse_wf,
fset_wf,
fset-singleton_wf,
empty-fset_wf,
bool_wf,
eqtt_to_assert,
member-fset-singleton,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
mem_empty_lemma,
assert_elim,
and_wf,
not_assert_elim,
btrue_neq_bfalse,
iff_weakening_uiff,
f-union_wf,
member-f-union,
fset-member_witness,
uiff_wf,
fset-constrained-image_wf,
deq_wf,
bool_cases,
assert_functionality_wrt_uiff,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
independent_pairFormation,
isect_memberFormation,
introduction,
sqequalHypSubstitution,
imageElimination,
productElimination,
thin,
dependent_pairFormation,
hypothesisEquality,
productEquality,
extract_by_obid,
isectElimination,
cumulativity,
hypothesis,
applyEquality,
functionExtensionality,
sqequalRule,
imageMemberEquality,
baseClosed,
lambdaEquality,
lambdaFormation,
unionElimination,
equalityElimination,
because_Cache,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
dependent_functionElimination,
instantiate,
independent_functionElimination,
voidElimination,
isect_memberEquality,
voidEquality,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
rename,
addLevel,
functionEquality,
universeEquality,
independent_pairEquality
Latex:
\mforall{}[T,A:Type]. \mforall{}[eqt:EqDecider(T)]. \mforall{}[eqa:EqDecider(A)]. \mforall{}[f:T {}\mrightarrow{} A]. \mforall{}[P:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[s:fset(T)].
\mforall{}[a:A].
uiff(a \mmember{} f"(s) s.t. P;\mdownarrow{}\mexists{}x:T. (x \mmember{} s \mwedge{} (a = (f x)) \mwedge{} (\muparrow{}(P a))))
Date html generated:
2017_04_17-AM-09_21_15
Last ObjectModification:
2017_02_27-PM-05_24_13
Theory : finite!sets
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