Nuprl Lemma : member-fpf-vals
∀[A:Type]
  ∀eq:EqDecider(A)
    ∀[B:A ⟶ Type]
      ∀P:A ⟶ 𝔹. ∀f:x:A fp-> B[x]. ∀x:A. ∀v:B[x].
        ((<x, v> ∈ fpf-vals(eq;P;f)) 
⇐⇒ {((↑x ∈ dom(f)) ∧ (↑(P x))) ∧ (v = f(x) ∈ B[x])})
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
, 
uall: ∀[x:A]. B[x]
, 
guard: {T}
, 
so_apply: x[s]
, 
all: ∀x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
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]
, 
all: ∀x:A. B[x]
, 
fpf: a:A fp-> B[a]
, 
fpf-ap: f(x)
, 
fpf-dom: x ∈ dom(f)
, 
fpf-vals: fpf-vals(eq;P;f)
, 
pi1: fst(t)
, 
pi2: snd(t)
, 
let: let, 
member: t ∈ T
, 
uimplies: b supposing a
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
implies: P 
⇒ Q
, 
rev_implies: P 
⇐ Q
, 
prop: ℙ
, 
sq_type: SQType(T)
, 
guard: {T}
, 
so_apply: x[s]
, 
so_lambda: λ2x.t[x]
, 
top: Top
, 
assert: ↑b
, 
ifthenelse: if b then t else f fi 
, 
bfalse: ff
, 
bnot: ¬bb
, 
istype: istype(T)
, 
uiff: uiff(P;Q)
, 
btrue: tt
, 
unit: Unit
, 
bool: 𝔹
, 
subtype_rel: A ⊆r B
, 
decidable: Dec(P)
, 
so_apply: x[s1;s2]
, 
so_lambda: λ2x y.t[x; y]
, 
squash: ↓T
, 
less_than: a < b
, 
it: ⋅
, 
nil: []
, 
colength: colength(L)
, 
less_than': less_than'(a;b)
, 
le: A ≤ B
, 
cons: [a / b]
, 
or: P ∨ Q
, 
exists: ∃x:A. B[x]
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
not: ¬A
, 
ge: i ≥ j 
, 
false: False
, 
nat: ℕ
, 
cand: A c∧ B
, 
deq: EqDecider(T)
, 
eqof: eqof(d)
, 
true: True
, 
label: ...$L... t
, 
rev_uimplies: rev_uimplies(P;Q)
, 
bor: p ∨bq
Lemmas referenced : 
subtype_base_sq, 
bool_wf, 
bool_subtype_base, 
iff_imp_equal_bool, 
deq-member_wf, 
remove-repeats_wf, 
member-remove-repeats, 
l_member_wf, 
assert-deq-member, 
istype-assert, 
fpf_wf, 
deq_wf, 
istype-universe, 
list_induction, 
filter_nil_lemma, 
istype-void, 
deq_member_nil_lemma, 
zip_nil_lemma, 
filter_cons_lemma, 
deq_member_cons_lemma, 
equal_wf, 
assert_wf, 
iff_wf, 
all_wf, 
nat_wf, 
assert-bnot, 
bool_cases_sqequal, 
eqff_to_assert, 
subtype_rel_sets, 
subtype_rel_dep_function, 
cons_member, 
cons_wf, 
zip_cons_cons_lemma, 
map_cons_lemma, 
eqtt_to_assert, 
decidable__le, 
int_term_value_add_lemma, 
int_term_value_subtract_lemma, 
int_formula_prop_not_lemma, 
itermAdd_wf, 
itermSubtract_wf, 
intformnot_wf, 
subtract_wf, 
decidable__equal_int, 
spread_cons_lemma, 
int_subtype_base, 
set_subtype_base, 
int_formula_prop_eq_lemma, 
intformeq_wf, 
subtract-1-ge-0, 
le_wf, 
istype-false, 
colength_wf_list, 
colength-cons-not-zero, 
product_subtype_list, 
nil_wf, 
list-cases, 
less_than_wf, 
ge_wf, 
int_formula_prop_wf, 
int_formula_prop_less_lemma, 
int_term_value_var_lemma, 
int_term_value_constant_lemma, 
int_formula_prop_le_lemma, 
int_formula_prop_and_lemma, 
istype-int, 
intformless_wf, 
itermVar_wf, 
itermConstant_wf, 
intformle_wf, 
intformand_wf, 
full-omega-unsat, 
nat_properties, 
false_wf, 
guard_wf, 
assert_witness, 
btrue_neq_bfalse, 
member-implies-null-eq-bfalse, 
btrue_wf, 
null_nil_lemma, 
list-subtype, 
subtype_rel_list, 
not_wf, 
bnot_wf, 
equal-wf-T-base, 
list_wf, 
set_wf, 
uiff_transitivity, 
assert_of_bnot, 
subtype_rel_product, 
bor_wf, 
iff_transitivity, 
eqof_wf, 
iff_weakening_uiff, 
assert_of_bor, 
safe-assert-deq, 
assert_elim, 
iff_weakening_equal, 
subtype_rel_self, 
true_wf, 
squash_wf, 
assert_functionality_wrt_uiff, 
or_wf, 
pi1_wf_top, 
subtype_rel-equal, 
not_assert_elim
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
lambdaFormation_alt, 
sqequalHypSubstitution, 
productElimination, 
thin, 
sqequalRule, 
cut, 
instantiate, 
introduction, 
extract_by_obid, 
isectElimination, 
cumulativity, 
hypothesis, 
independent_isectElimination, 
hypothesisEquality, 
independent_pairFormation, 
dependent_functionElimination, 
independent_functionElimination, 
universeIsType, 
because_Cache, 
promote_hyp, 
equalityTransitivity, 
equalitySymmetry, 
inhabitedIsType, 
equalityIstype, 
applyEquality, 
lambdaEquality_alt, 
functionIsType, 
universeEquality, 
setIsType, 
dependent_set_memberEquality_alt, 
rename, 
setElimination, 
functionExtensionality_alt, 
applyLambdaEquality, 
hyp_replacement, 
isect_memberEquality_alt, 
voidElimination, 
dependent_pairEquality_alt, 
productEquality, 
setEquality, 
functionEquality, 
inrFormation_alt, 
inlFormation_alt, 
equalityElimination, 
intEquality, 
baseClosed, 
closedConclusion, 
baseApply, 
equalityIsType4, 
imageElimination, 
equalityIsType1, 
hypothesis_subsumption, 
unionElimination, 
functionIsTypeImplies, 
axiomEquality, 
int_eqEquality, 
dependent_pairFormation_alt, 
approximateComputation, 
natural_numberEquality, 
intWeakElimination, 
independent_pairEquality, 
unionEquality, 
unionIsType, 
imageMemberEquality, 
productIsType
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.  \mforall{}v:B[x].
                ((<x,  v>  \mmember{}  fpf-vals(eq;P;f))  \mLeftarrow{}{}\mRightarrow{}  \{((\muparrow{}x  \mmember{}  dom(f))  \mwedge{}  (\muparrow{}(P  x)))  \mwedge{}  (v  =  f(x))\})
Date html generated:
2019_10_16-AM-11_26_03
Last ObjectModification:
2019_06_25-PM-03_26_35
Theory : finite!partial!functions
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