Nuprl Lemma : finite-function
∀S:Type. ∀T:S ⟶ Type. (finite(S) ⇒ (∀s:S. finite(T[s])) ⇒ finite(s:S ⟶ T[s]))
Proof
Definitions occuring in Statement :
finite: finite(T),
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
finite: finite(T),
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
subtype_rel: A ⊆r B,
pi1: fst(t),
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
guard: {T},
equipollent: A ~ B,
compose: f o g,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
surject: Surj(A;B;f),
squash: ↓T,
true: True,
inv_funs: InvFuns(A;B;f;g),
tidentity: Id{T},
identity: Id,
int_seg: {i..j-},
sq_type: SQType(T)
Lemmas referenced :
all_wf,
finite_wf,
exists_wf,
nat_wf,
equipollent_wf,
int_seg_wf,
equal_wf,
function_functionality_wrt_equipollent,
equipollent_functionality_wrt_equipollent,
equipollent_weakening_ext-eq,
ext-eq_weakening,
equipollent_inversion,
equipollent-product,
compose_wf,
int-prod_wf_nat,
int-prod_wf,
equipollent_functionality_wrt_equipollent2,
biject_wf,
squash_wf,
true_wf,
iff_weakening_equal,
bij_imp_exists_inv,
subtype_base_sq,
set_subtype_base,
lelt_wf,
int_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
sqequalRule,
productElimination,
thin,
cut,
hypothesis,
promote_hyp,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
functionEquality,
universeEquality,
because_Cache,
natural_numberEquality,
setElimination,
rename,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
addLevel,
existsFunctionality,
independent_isectElimination,
independent_pairFormation,
applyLambdaEquality,
hyp_replacement,
imageElimination,
instantiate,
intEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}S:Type. \mforall{}T:S {}\mrightarrow{} Type. (finite(S) {}\mRightarrow{} (\mforall{}s:S. finite(T[s])) {}\mRightarrow{} finite(s:S {}\mrightarrow{} T[s]))
Date html generated:
2017_04_17-AM-09_33_56
Last ObjectModification:
2017_02_27-PM-05_33_08
Theory : equipollence!!cardinality!
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