Nuprl Lemma : member-fpf-vals2
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[B:A ─→ Type]. ∀[P:A ─→ 𝔹]. ∀[f:x:A fp-> B[x]]. ∀[x:{a:A| ↑(P a)} ]. ∀[v:B[x]].
{(↑x ∈ dom(f)) ∧ (v = f(x) ∈ B[x])} supposing (<x, v> ∈ fpf-vals(eq;P;f))
Proof
Definitions occuring in Statement :
fpf-vals: fpf-vals(eq;P;f),
fpf-ap: f(x),
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
deq: EqDecider(T),
l_member: (x ∈ l),
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
so_apply: x[s],
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ─→ B[x],
pair: <a, b>,
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Lemmas :
member-fpf-vals,
l_member_wf,
assert_wf,
fpf-vals_wf,
set_wf,
fpf_wf,
bool_wf,
deq_wf,
l_member_subtype,
subtype_rel_product,
subtype_rel_self,
assert_witness,
fpf-dom_wf,
subtype-fpf2,
top_wf,
subtype_top
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[P:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[f:x:A fp-> B[x]]. \mforall{}[x:\{a:A| \muparrow{}(P a)\} ].
\mforall{}[v:B[x]].
\{(\muparrow{}x \mmember{} dom(f)) \mwedge{} (v = f(x))\} supposing (<x, v> \mmember{} fpf-vals(eq;P;f))
Date html generated:
2015_07_17-AM-11_09_30
Last ObjectModification:
2015_01_28-AM-07_44_54
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