Nuprl Lemma : equipollent-quotient2
∀[A:Type]
∀E:A ⟶ A ⟶ ℙ. ∀d:∀x,y:A. Dec(↓E[x;y]).
A ~ a:x,y:A//(↓E[x;y]) × {b:A| ↑isl(d 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,
isl: isl(x),
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
squash: ↓T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
so_lambda: λ2x y.t[x; y],
not: ¬A,
false: False,
bfalse: ff,
true: True,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
isl: isl(x),
or: P ∨ Q,
decidable: Dec(P),
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
trans: Trans(T;x,y.E[x; y]),
prop: ℙ,
so_apply: x[s1;s2],
implies: P ⇒ Q,
sym: Sym(T;x,y.E[x; y]),
squash: ↓T,
refl: Refl(T;x,y.E[x; y]),
and: P ∧ Q,
equiv_rel: EquivRel(T;x,y.E[x; y]),
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
quotient: x,y:A//B[x; y],
ext-eq: A ≡ B,
guard: {T},
equiv-class: equiv-class(A;a,b.E[a; b];t)
Lemmas referenced :
equiv_rel_wf,
equipollent-quotient,
assert_witness,
assert_wf,
false_wf,
true_wf,
equal_wf,
not_wf,
isl_wf,
decidable_wf,
all_wf,
squash_wf,
equiv_rel_functionality_wrt_iff,
quotient_wf,
equal-wf-base,
quotient-member-eq,
subtype_rel_weakening,
equiv-class_wf,
equipollent_functionality_wrt_equipollent,
equipollent_weakening_ext-eq,
ext-eq_weakening,
ext-eq_inversion,
product_functionality_wrt_equipollent_dependent,
ext-eq-implies-biject
Rules used in proof :
universeEquality,
functionEquality,
independent_isectElimination,
because_Cache,
voidElimination,
natural_numberEquality,
independent_pairFormation,
unionElimination,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
rename,
cumulativity,
functionExtensionality,
applyEquality,
isectElimination,
extract_by_obid,
baseClosed,
imageMemberEquality,
hypothesis,
imageElimination,
hypothesisEquality,
dependent_functionElimination,
lambdaEquality,
independent_pairEquality,
thin,
productElimination,
sqequalHypSubstitution,
sqequalRule,
introduction,
cut,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productEquality,
pertypeElimination,
pointwiseFunctionalityForEquality
Latex:
\mforall{}[A:Type]
\mforall{}E:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}. \mforall{}d:\mforall{}x,y:A. Dec(\mdownarrow{}E[x;y]).
A \msim{} a:x,y:A//(\mdownarrow{}E[x;y]) \mtimes{} \{b:A| \muparrow{}isl(d a b)\} supposing EquivRel(A;x,y.\mdownarrow{}E[x;y])
Date html generated:
2018_05_21-PM-00_52_53
Last ObjectModification:
2018_01_05-AM-10_24_53
Theory : equipollence!!cardinality!
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