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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