Nuprl Lemma : finite-injective-quotient
∀T,S:Type. ∀f:T ⟶ S. (finite(S) ⇒ (∀s:S. Dec(∃t:T. (f[t] = s ∈ S))) ⇒ finite(T//t.f[t]))
Proof
Definitions occuring in Statement :
finite: finite(T),
injective-quotient: T//x.f[x],
decidable: Dec(P),
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
exists: ∃x:A. B[x],
so_apply: x[s],
injective-quotient: T//x.f[x],
quotient: x,y:A//B[x; y],
and: P ∧ Q,
so_lambda: λ2x.t[x],
equipollent: A ~ B,
guard: {T},
biject: Bij(A;B;f),
inject: Inj(A;B;f),
surject: Surj(A;B;f),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
equiv_rel: EquivRel(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
squash: ↓T,
sq_stable: SqStable(P),
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
decidable_wf,
equal_wf,
finite_wf,
istype-universe,
injective-quotient_wf,
biject_wf,
quotient-member-eq,
sq_stable_from_decidable,
subtype_quotient,
finite_functionality_wrt_equipollent,
exists_wf,
finite-decidable-subset,
decidable__squash
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
sqequalRule,
Error :functionIsType,
Error :universeIsType,
hypothesisEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
productEquality,
applyEquality,
hypothesis,
Error :inhabitedIsType,
instantiate,
universeEquality,
functionExtensionality,
pointwiseFunctionalityForEquality,
setEquality,
pertypeElimination,
promote_hyp,
productElimination,
Error :dependent_set_memberEquality_alt,
Error :dependent_pairFormation_alt,
equalityTransitivity,
equalitySymmetry,
Error :equalityIstype,
independent_functionElimination,
Error :productIsType,
because_Cache,
sqequalBase,
Error :lambdaEquality_alt,
independent_pairFormation,
Error :setIsType,
independent_isectElimination,
dependent_functionElimination,
Error :equalityIsType4,
Error :equalityIsType1,
applyLambdaEquality,
setElimination,
rename,
imageMemberEquality,
baseClosed,
imageElimination,
cumulativity
Latex:
\mforall{}T,S:Type. \mforall{}f:T {}\mrightarrow{} S. (finite(S) {}\mRightarrow{} (\mforall{}s:S. Dec(\mexists{}t:T. (f[t] = s))) {}\mRightarrow{} finite(T//t.f[t]))
Date html generated:
2019_06_20-PM-02_19_14
Last ObjectModification:
2018_12_16-PM-00_26_12
Theory : equipollence!!cardinality!
Home
Index