Nuprl Lemma : subtype_quotient
∀[T:Type]. ∀[E:T ⟶ T ⟶ ℙ]. T ⊆r (x,y:T//E[x;y]) supposing EquivRel(T;x,y.E[x;y])
Proof
Definitions occuring in Statement :
equiv_rel: EquivRel(T;x,y.E[x; y]),
quotient: x,y:A//B[x; y],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
quotient: x,y:A//B[x; y],
and: P ∧ Q,
cand: A c∧ B,
guard: {T},
equiv_rel: EquivRel(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
all: ∀x:A. B[x],
squash: ↓T,
true: True
Lemmas referenced :
equiv_rel_wf,
quotient_wf,
member_wf,
squash_wf,
true_wf,
and_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaEquality,
pointwiseFunctionality,
hypothesisEquality,
sqequalRule,
axiomEquality,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
applyEquality,
functionExtensionality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
independent_isectElimination,
independent_pairFormation,
pertypeMemberEquality,
productElimination,
dependent_functionElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
dependent_set_memberEquality,
setElimination,
rename,
setEquality
Latex:
\mforall{}[T:Type]. \mforall{}[E:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. T \msubseteq{}r (x,y:T//E[x;y]) supposing EquivRel(T;x,y.E[x;y])
Date html generated:
2016_10_21-AM-09_43_42
Last ObjectModification:
2016_07_12-AM-05_04_19
Theory : quot_1
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