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],
l_member: (x ∈ l),
deq: EqDecider(T),
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
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
guard: {T},
top: Top
Lemmas referenced :
member-fpf-vals,
l_member_wf,
assert_wf,
fpf-vals_wf,
fpf_wf,
bool_wf,
deq_wf,
l_member_subtype,
subtype_rel_product,
subtype_rel_self,
set_wf,
assert_witness,
fpf-dom_wf,
subtype-fpf2,
top_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
setElimination,
rename,
productElimination,
independent_functionElimination,
productEquality,
setEquality,
applyEquality,
hypothesis,
dependent_pairEquality,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
universeEquality,
independent_isectElimination,
because_Cache,
lambdaFormation,
independent_pairEquality,
isect_memberEquality,
voidElimination,
voidEquality,
axiomEquality
Latex:
\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:
2018_05_21-PM-09_25_50
Last ObjectModification:
2018_02_09-AM-10_21_31
Theory : finite!partial!functions
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