Nuprl Lemma : fset-image-add
∀[T,A:Type]. ∀[eqt:EqDecider(T)]. ∀[eqa:EqDecider(A)]. ∀[f:T ⟶ A]. ∀[x:T]. ∀[s:fset(T)].
(f"(fset-add(eqt;x;s)) = {f x} ⋃ f"(s) ∈ fset(A))
Proof
Definitions occuring in Statement :
fset-image: f"(s),
fset-add: fset-add(eq;x;s),
fset-singleton: {x},
fset-union: x ⋃ y,
fset: fset(T),
deq: EqDecider(T),
uall: ∀[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],
member: t ∈ T,
top: Top,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
fset-add: fset-add(eq;x;s)
Lemmas referenced :
fset-image-singleton,
fset-singleton_wf,
equal_wf,
squash_wf,
true_wf,
fset_wf,
fset-image_wf,
fset-add_wf,
fset-image-union,
iff_weakening_equal,
fset-union_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
cumulativity,
hypothesisEquality,
applyEquality,
functionExtensionality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
productElimination,
independent_functionElimination,
hyp_replacement,
applyLambdaEquality,
axiomEquality,
functionEquality
Latex:
\mforall{}[T,A:Type]. \mforall{}[eqt:EqDecider(T)]. \mforall{}[eqa:EqDecider(A)]. \mforall{}[f:T {}\mrightarrow{} A]. \mforall{}[x:T]. \mforall{}[s:fset(T)].
(f"(fset-add(eqt;x;s)) = \{f x\} \mcup{} f"(s))
Date html generated:
2017_04_17-AM-09_20_57
Last ObjectModification:
2017_02_27-PM-05_23_55
Theory : finite!sets
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