Nuprl Lemma : regextfun_wf2
∀[T:Type]. ∀[f:T ⟶ Set{i:l}]. ∀[w:W(T;x.set-dom(f x))].  (regextfun(f;w) ∈ Set{i:l})
Proof
Definitions occuring in Statement : 
regextfun: regextfun(f;w)
, 
Set: Set{i:l}
, 
set-dom: set-dom(s)
, 
W: W(A;a.B[a])
, 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
apply: f a
, 
function: x:A ⟶ B[x]
, 
universe: Type
Definitions unfolded in proof : 
sq_stable: SqStable(P)
, 
le: A ≤ B
, 
sq_type: SQType(T)
, 
cand: A c∧ B
, 
pcw-step-agree: StepAgree(s;p1;w)
, 
isl: isl(x)
, 
pi2: snd(t)
, 
pcw-steprel: StepRel(s1;s2)
, 
param-W-rel: param-W-rel(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b];par;w)
, 
W-rel: W-rel(A;a.B[a];w)
, 
nat_plus: ℕ+
, 
pi1: fst(t)
, 
ext-family: F ≡ G
, 
so_apply: x[s1;s2;s3]
, 
so_lambda: so_lambda(x,y,z.t[x; y; z])
, 
so_apply: x[s1;s2]
, 
so_lambda: λ2x y.t[x; y]
, 
it: ⋅
, 
unit: Unit
, 
ext-eq: A ≡ B
, 
btrue: tt
, 
bfalse: ff
, 
ifthenelse: if b then t else f fi 
, 
assert: ↑b
, 
isr: isr(x)
, 
squash: ↓T
, 
true: True
, 
less_than': less_than'(a;b)
, 
less_than: a < b
, 
spreadn: spread3, 
pcw-step: pcw-step(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b])
, 
cw-step: cw-step(A;a.B[a])
, 
top: Top
, 
false: False
, 
exists: ∃x:A. B[x]
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
not: ¬A
, 
uimplies: b supposing a
, 
or: P ∨ Q
, 
decidable: Dec(P)
, 
ge: i ≥ j 
, 
lelt: i ≤ j < k
, 
int_seg: {i..j-}
, 
nat: ℕ
, 
pcw-pp-barred: Barred(pp)
, 
implies: P 
⇒ Q
, 
prop: ℙ
, 
and: P ∧ Q
, 
guard: {T}
, 
all: ∀x:A. B[x]
, 
Wsup: Wsup(a;b)
, 
regextfun: regextfun(f;w)
, 
so_apply: x[s]
, 
subtype_rel: A ⊆r B
, 
so_lambda: λ2x.t[x]
, 
member: t ∈ T
, 
uall: ∀[x:A]. B[x]
Lemmas referenced : 
sq_stable__le, 
int_seg_subtype, 
subtype_rel_function, 
int_formula_prop_eq_lemma, 
intformeq_wf, 
decidable__equal_int, 
int_subtype_base, 
le_wf, 
set_subtype_base, 
nat_wf, 
subtype_base_sq, 
subtype_rel_dep_function, 
pcw-steprel_wf, 
param-co-W_wf, 
it_wf, 
unit_wf2, 
param-co-W-ext, 
W-ext, 
int_term_value_add_lemma, 
itermAdd_wf, 
add-subtract-cancel, 
equal_wf, 
true_wf, 
false_wf, 
less_than_wf, 
top_wf, 
lelt_wf, 
decidable__lt, 
int_formula_prop_wf, 
int_formula_prop_less_lemma, 
int_term_value_var_lemma, 
int_term_value_subtract_lemma, 
int_term_value_constant_lemma, 
int_formula_prop_le_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_and_lemma, 
intformless_wf, 
itermVar_wf, 
itermSubtract_wf, 
itermConstant_wf, 
intformle_wf, 
intformnot_wf, 
intformand_wf, 
full-omega-unsat, 
decidable__le, 
nat_properties, 
subtract_wf, 
int_seg_wf, 
subtype_rel_self, 
W-elimination-facts, 
mk-set_wf, 
Set_wf, 
set-subtype-coSet, 
set-dom_wf, 
W_wf
Rules used in proof : 
applyLambdaEquality, 
hyp_replacement, 
productEquality, 
unionEquality, 
inlEquality, 
dependent_pairEquality, 
equalityElimination, 
hypothesis_subsumption, 
promote_hyp, 
int_eqReduceTrueSq, 
addEquality, 
axiomEquality, 
imageElimination, 
baseClosed, 
imageMemberEquality, 
sqequalAxiom, 
lessCases, 
lambdaFormation, 
voidEquality, 
voidElimination, 
isect_memberEquality, 
intEquality, 
int_eqEquality, 
dependent_pairFormation, 
approximateComputation, 
independent_isectElimination, 
unionElimination, 
independent_pairFormation, 
dependent_set_memberEquality, 
rename, 
setElimination, 
natural_numberEquality, 
functionExtensionality, 
independent_functionElimination, 
instantiate, 
equalitySymmetry, 
equalityTransitivity, 
strong_bar_Induction, 
productElimination, 
dependent_functionElimination, 
because_Cache, 
universeEquality, 
cumulativity, 
functionEquality, 
applyEquality, 
lambdaEquality, 
sqequalRule, 
hypothesisEquality, 
thin, 
isectElimination, 
extract_by_obid, 
introduction, 
hypothesis, 
sqequalHypSubstitution, 
cut, 
isect_memberFormation, 
sqequalReflexivity, 
computationStep, 
sqequalTransitivity, 
sqequalSubstitution
Latex:
\mforall{}[T:Type].  \mforall{}[f:T  {}\mrightarrow{}  Set\{i:l\}].  \mforall{}[w:W(T;x.set-dom(f  x))].    (regextfun(f;w)  \mmember{}  Set\{i:l\})
Date html generated:
2018_07_29-AM-10_07_08
Last ObjectModification:
2018_07_20-PM-04_50_55
Theory : constructive!set!theory
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