Nuprl Lemma : equiv-equipollent-iff-quotient-equipollent
∀[A,B:Type].
∀E:A ⟶ A ⟶ ℙ
((∃g:(x,y:A//E[x;y]) ⟶ A. ∀c:x,y:A//E[x;y]. ((g c) = c ∈ (x,y:A//E[x;y])))
∨ (∀f:A ⟶ B. ∀b:B. SqStable(∃a:A. ((f a) = b ∈ B))))
⇒ (A ~ B mod (a1,a2.E[a1;a2]) ⇐⇒ x,y:A//E[x;y] ~ B)
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,
equiv_rel: EquivRel(T;x,y.E[x; y]),
quotient: x,y:A//B[x; y],
sq_stable: SqStable(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
or: P ∨ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
prop: ℙ,
rev_implies: P ⇐ Q,
or: P ∨ Q,
exists: ∃x:A. B[x],
respects-equality: respects-equality(S;T),
equipollent: A ~ B,
biject: Bij(A;B;f),
equiv-equipollent: A ~ B mod (a1,a2.E[a1; a2]),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B,
squash: ↓T,
surject: Surj(A;B;f),
true: True,
guard: {T},
quotient: x,y:A//B[x; y],
inject: Inj(A;B;f),
sq_stable: SqStable(P)
Lemmas referenced :
equiv-equipollent-implies-quotient-equipollent,
equiv-equipollent_wf,
equipollent_wf,
quotient_wf,
subtype-respects-equality,
subtype_quotient,
sq_stable_wf,
equal_wf,
equiv_rel_wf,
istype-universe,
subtype_rel_dep_function,
squash_wf,
surject_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
quotient-member-eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
dependent_functionElimination,
sqequalRule,
Error :lambdaEquality_alt,
independent_isectElimination,
hypothesis,
independent_functionElimination,
Error :universeIsType,
applyEquality,
Error :inhabitedIsType,
unionElimination,
productElimination,
Error :unionIsType,
Error :productIsType,
Error :functionIsType,
Error :equalityIstype,
productEquality,
universeEquality,
instantiate,
Error :dependent_pairFormation_alt,
imageElimination,
imageMemberEquality,
baseClosed,
independent_pairEquality,
Error :functionIsTypeImplies,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
pertypeElimination,
pointwiseFunctionality,
promote_hyp,
sqequalBase
Latex:
\mforall{}[A,B:Type].
\mforall{}E:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}
((\mexists{}g:(x,y:A//E[x;y]) {}\mrightarrow{} A. \mforall{}c:x,y:A//E[x;y]. ((g c) = c))
\mvee{} (\mforall{}f:A {}\mrightarrow{} B. \mforall{}b:B. SqStable(\mexists{}a:A. ((f a) = b))))
{}\mRightarrow{} (A \msim{} B mod (a1,a2.E[a1;a2]) \mLeftarrow{}{}\mRightarrow{} x,y:A//E[x;y] \msim{} B)
supposing EquivRel(A;x,y.E[x;y])
Date html generated:
2019_06_20-PM-02_17_45
Last ObjectModification:
2018_11_25-PM-01_10_41
Theory : equipollence!!cardinality!
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