Nuprl Lemma : equiv_rel_subtype
∀[T,S:Type]. ∀[R:T ⟶ T ⟶ Type]. EquivRel(T;x,y.R[x;y]) ⇒ EquivRel(S;x,y.R[x;y]) supposing S ⊆r T
Proof
Definitions occuring in Statement :
equiv_rel: EquivRel(T;x,y.E[x; y]),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
equiv_rel: EquivRel(T;x,y.E[x; y]),
and: P ∧ Q,
refl: Refl(T;x,y.E[x; y]),
all: ∀x:A. B[x],
sym: Sym(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
equiv_rel_wf,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
sqequalRule,
axiomEquality,
hypothesis,
thin,
rename,
lambdaFormation,
sqequalHypSubstitution,
independent_pairFormation,
productElimination,
promote_hyp,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
because_Cache,
lemma_by_obid,
isectElimination,
lambdaEquality,
universeEquality,
functionEquality,
cumulativity
Latex:
\mforall{}[T,S:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} Type].
EquivRel(T;x,y.R[x;y]) {}\mRightarrow{} EquivRel(S;x,y.R[x;y]) supposing S \msubseteq{}r T
Date html generated:
2016_05_13-PM-04_15_02
Last ObjectModification:
2015_12_26-AM-11_30_05
Theory : rel_1
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