Nuprl Lemma : equiv-equipollent-implies-quotient-equipollent
∀[A,B:Type]. ∀E:A ⟶ A ⟶ ℙ. 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],
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
sq_stable: SqStable(P),
equiv_rel: EquivRel(T;x,y.E[x; y]),
and: P ∧ Q,
refl: Refl(T;x,y.E[x; y]),
squash: ↓T,
cand: A c∧ B,
sym: Sym(T;x,y.E[x; y]),
prop: ℙ,
trans: Trans(T;x,y.E[x; y]),
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
equiv-equipollent: A ~ B mod (a1,a2.E[a1; a2]),
exists: ∃x:A. B[x],
quotient: x,y:A//B[x; y],
equipollent: A ~ B,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
surject: Surj(A;B;f),
subtype_rel: A ⊆r B
Lemmas referenced :
equipollent_weakening_ext-eq,
quotient_wf,
squash_wf,
equiv_rel_squash,
quotient-squash,
equiv_rel_wf,
equiv-equipollent_wf,
equipollent_functionality_wrt_equipollent,
ext-eq_weakening,
equal-wf-base,
biject_wf,
quotient-member-eq,
subtype_quotient
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
Error :lambdaEquality_alt,
applyEquality,
functionExtensionality,
because_Cache,
Error :inhabitedIsType,
Error :universeIsType,
independent_isectElimination,
hypothesis,
independent_functionElimination,
independent_pairFormation,
imageElimination,
imageMemberEquality,
baseClosed,
productElimination,
dependent_functionElimination,
Error :functionIsType,
universeEquality,
pointwiseFunctionalityForEquality,
pertypeElimination,
equalityTransitivity,
equalitySymmetry,
productEquality,
Error :dependent_pairFormation_alt,
Error :equalityIsType1,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
applyLambdaEquality,
setElimination,
rename
Latex:
\mforall{}[A,B:Type].
\mforall{}E:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}. A \msim{} B mod (a1,a2.E[a1;a2]) {}\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_39
Last ObjectModification:
2018_10_02-PM-11_35_29
Theory : equipollence!!cardinality!
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