Nuprl Lemma : type_inj_wf_for_quot
∀[T:Type]. ∀[E:T ⟶ T ⟶ ℙ]. ∀[a:T]. ([a]{x,y:T//E[x;y]} ∈ 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],
type_inj: [x]{T},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
type_inj: [x]{T},
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
prop: ℙ
Lemmas referenced :
subtype_quotient,
equiv_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
hypothesisEquality,
applyEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
lambdaEquality,
independent_isectElimination,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
because_Cache,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[E:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
\mforall{}[a:T]. ([a]\{x,y:T//E[x;y]\} \mmember{} x,y:T//E[x;y]) supposing EquivRel(T;x,y.E[x;y])
Date html generated:
2019_06_20-PM-00_32_08
Last ObjectModification:
2018_09_17-PM-07_05_00
Theory : quot_1
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