Nuprl Lemma : member-fset-image-iff
∀[T,A:Type]. ∀[eqt:EqDecider(T)]. ∀[eqa:EqDecider(A)]. ∀[f:T ⟶ A]. ∀[s:fset(T)]. ∀[a:A].
uiff(a ∈ f"(s);↓∃x:T. (x ∈ s ∧ (a = (f x) ∈ A)))
Proof
Definitions occuring in Statement :
fset-image: f"(s),
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
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-image: f"(s),
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
uiff: uiff(P;Q),
uimplies: b supposing a,
squash: ↓T,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
fset-member_wf,
fset-image_wf,
squash_wf,
exists_wf,
equal_wf,
fset_wf,
deq_wf,
fset-member_witness,
member-fset-singleton,
fset-singleton_wf,
uiff_wf,
iff_weakening_uiff,
f-union_wf,
member-f-union
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
functionExtensionality,
applyEquality,
hypothesis,
sqequalRule,
lambdaEquality,
productEquality,
because_Cache,
functionEquality,
universeEquality,
isect_memberFormation,
productElimination,
independent_pairEquality,
isect_memberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
addLevel,
independent_pairFormation,
independent_isectElimination,
existsFunctionality,
andLevelFunctionality
Latex:
\mforall{}[T,A:Type]. \mforall{}[eqt:EqDecider(T)]. \mforall{}[eqa:EqDecider(A)]. \mforall{}[f:T {}\mrightarrow{} A]. \mforall{}[s:fset(T)]. \mforall{}[a:A].
uiff(a \mmember{} f"(s);\mdownarrow{}\mexists{}x:T. (x \mmember{} s \mwedge{} (a = (f x))))
Date html generated:
2017_04_17-AM-09_20_50
Last ObjectModification:
2017_02_27-PM-05_24_05
Theory : finite!sets
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