Nuprl Lemma : equiv-class_wf
∀[A:Type]. ∀[E:A ⟶ A ⟶ 𝔹].
∀[t:x,y:A//(↑E[x;y])]. (equiv-class(A;x,y.E[x;y];t) ∈ Type) supposing EquivRel(A;x,y.↑E[x;y])
Proof
Definitions occuring in Statement :
equiv-class: equiv-class(A;a,b.E[a; b];t),
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],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
quotient: x,y:A//B[x; y],
and: P ∧ Q,
equiv-class: equiv-class(A;a,b.E[a; b];t),
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
equiv_rel: EquivRel(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
all: ∀x:A. B[x],
sym: Sym(T;x,y.E[x; y])
Lemmas referenced :
assert_wf,
squash_wf,
true_wf,
bool_wf,
equal-wf-base,
quotient_wf,
equiv_rel_wf,
iff_imp_equal_bool
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
pointwiseFunctionalityForEquality,
universeEquality,
sqequalHypSubstitution,
sqequalRule,
pertypeElimination,
productElimination,
thin,
setEquality,
hypothesisEquality,
applyEquality,
lambdaEquality,
imageElimination,
extract_by_obid,
isectElimination,
equalityTransitivity,
hypothesis,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productEquality,
because_Cache,
functionExtensionality,
cumulativity,
axiomEquality,
independent_isectElimination,
isect_memberEquality,
functionEquality,
independent_pairFormation,
lambdaFormation,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[A:Type]. \mforall{}[E:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbB{}].
\mforall{}[t:x,y:A//(\muparrow{}E[x;y])]. (equiv-class(A;x,y.E[x;y];t) \mmember{} Type) supposing EquivRel(A;x,y.\muparrow{}E[x;y])
Date html generated:
2016_10_21-AM-09_43_50
Last ObjectModification:
2016_08_07-PM-06_00_35
Theory : quot_1
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