Nuprl Lemma : equipollent-quotient
∀[A:Type]. ∀E:A ⟶ A ⟶ 𝔹. A ~ a:x,y:A//(↑E[x;y]) × {b:A| ↑E[a;b]} supposing EquivRel(A;x,y.↑E[x;y])
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
equiv_rel: EquivRel(T;x,y.E[x; y]),
quotient: x,y:A//B[x; y],
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
equiv_rel: EquivRel(T;x,y.E[x; y]),
and: P ∧ Q,
refl: Refl(T;x,y.E[x; y]),
so_apply: x[s1;s2],
implies: P ⇒ Q,
sym: Sym(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
quotient: x,y:A//B[x; y],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x y.t[x; y],
prop: ℙ,
guard: {T},
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
sq_type: SQType(T),
equipollent: A ~ B,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
surject: Surj(A;B;f),
pi2: snd(t),
squash: ↓T,
istype: istype(T)
Lemmas referenced :
assert_witness,
bool_wf,
iff_imp_equal_bool,
istype-assert,
quotient_wf,
assert_wf,
equiv_rel_wf,
istype-universe,
btrue_wf,
true_wf,
subtype_base_sq,
bool_subtype_base,
subtype_quotient,
equal_wf,
squash_wf,
quotient-member-eq,
subtype_rel_self,
biject_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
Error :lambdaEquality_alt,
dependent_functionElimination,
hypothesisEquality,
extract_by_obid,
isectElimination,
applyEquality,
independent_functionElimination,
hypothesis,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
rename,
Error :functionExtensionality_alt,
pointwiseFunctionalityForEquality,
functionEquality,
pertypeElimination,
promote_hyp,
because_Cache,
independent_isectElimination,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
Error :productIsType,
Error :equalityIstype,
sqequalBase,
Error :universeIsType,
Error :functionIsType,
instantiate,
universeEquality,
natural_numberEquality,
cumulativity,
Error :dependent_pairFormation_alt,
Error :setIsType,
Error :dependent_set_memberEquality_alt,
Error :dependent_pairEquality_alt,
applyLambdaEquality,
setElimination,
pointwiseFunctionality,
Error :equalityIsType4,
imageElimination,
imageMemberEquality,
baseClosed,
productEquality,
setEquality
Latex:
\mforall{}[A:Type]. \mforall{}E:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbB{}. A \msim{} a:x,y:A//(\muparrow{}E[x;y]) \mtimes{} \{b:A| \muparrow{}E[a;b]\} supposing EquivRel(A;x,y.\muparrow{}E[x;y]\000C)
Date html generated:
2019_06_20-PM-02_17_35
Last ObjectModification:
2018_11_25-PM-01_28_18
Theory : equipollence!!cardinality!
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