Nuprl Lemma : quotient-equipollent
∀[A,B:Type].
(finite-type(A)
⇒ (∀x,y:B. Dec(x = y ∈ B))
⇒ (∀E:A ⟶ A ⟶ ℙ. x,y:A//E[x;y] ~ B ⇐⇒ A ~ B mod (a1,a2.E[a1;a2]) supposing EquivRel(A;x,y.E[x;y])))
Proof
Definitions occuring in Statement :
equiv-equipollent: A ~ B mod (a1,a2.E[a1; a2]),
equipollent: A ~ B,
finite-type: finite-type(T),
equiv_rel: EquivRel(T;x,y.E[x; y]),
quotient: x,y:A//B[x; y],
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
rev_implies: P ⇐ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
or: P ∨ Q,
subtype_rel: A ⊆r B
Lemmas referenced :
equiv-equipollent-iff-quotient-equipollent,
equipollent_wf,
quotient_wf,
equiv-equipollent_wf,
equiv_rel_wf,
all_wf,
decidable_wf,
equal_wf,
finite-type_wf,
sq_stable-finite-type-onto,
exists_wf,
subtype_quotient
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
dependent_functionElimination,
independent_isectElimination,
independent_functionElimination,
productElimination,
independent_pairFormation,
sqequalRule,
lambdaEquality,
applyEquality,
functionEquality,
cumulativity,
universeEquality,
inrFormation
Latex:
\mforall{}[A,B:Type].
(finite-type(A)
{}\mRightarrow{} (\mforall{}x,y:B. Dec(x = y))
{}\mRightarrow{} (\mforall{}E:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}
x,y:A//E[x;y] \msim{} B \mLeftarrow{}{}\mRightarrow{} A \msim{} B mod (a1,a2.E[a1;a2]) supposing EquivRel(A;x,y.E[x;y])))
Date html generated:
2018_05_21-PM-00_53_10
Last ObjectModification:
2018_05_19-AM-06_40_05
Theory : equipollence!!cardinality!
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