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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