Nuprl Lemma : f-union_wf
∀[T,A:Type]. ∀[eqt:EqDecider(T)]. ∀[eqa:EqDecider(A)]. ∀[g:T ⟶ fset(A)]. ∀[s:fset(T)].
(f-union(eqt;eqa;s;x.g[x]) ∈ fset(A))
Proof
Definitions occuring in Statement :
f-union: f-union(domeq;rngeq;s;x.g[x]),
fset: fset(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
fset: fset(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
f-union: f-union(domeq;rngeq;s;x.g[x]),
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
all: ∀x:A. B[x],
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
comm: Comm(T;op),
infix_ap: x f y,
assoc: Assoc(T;op)
Lemmas referenced :
fset_wf,
equal-wf-base,
list_wf,
set-equal_wf,
deq_wf,
list_accum-set-equal-idemp,
fset-union_wf,
equal_wf,
squash_wf,
true_wf,
fset-union-idempotent,
iff_weakening_equal,
nil_wf,
list_subtype_fset,
fset-union-commutes,
fset-union-associative
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
extract_by_obid,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
pertypeElimination,
productElimination,
productEquality,
because_Cache,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
functionEquality,
universeEquality,
lambdaEquality,
independent_isectElimination,
lambdaFormation,
applyEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
voidEquality,
voidElimination
Latex:
\mforall{}[T,A:Type]. \mforall{}[eqt:EqDecider(T)]. \mforall{}[eqa:EqDecider(A)]. \mforall{}[g:T {}\mrightarrow{} fset(A)]. \mforall{}[s:fset(T)].
(f-union(eqt;eqa;s;x.g[x]) \mmember{} fset(A))
Date html generated:
2017_04_17-AM-09_19_05
Last ObjectModification:
2017_02_27-PM-05_22_32
Theory : finite!sets
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