Nuprl Lemma : equiv_rel_subtyping
∀[T:Type]. ∀[R:T ⟶ T ⟶ Type]. ∀[Q:T ⟶ ℙ]. (EquivRel(T;x,y.R[x;y]) ⇒ EquivRel({z:T| Q[z]} ;x,y.R[x;y]))
Proof
Definitions occuring in Statement :
equiv_rel: EquivRel(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
member: t ∈ T,
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a
Lemmas referenced :
equiv_rel_subtype,
equiv_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lambdaEquality,
setElimination,
thin,
rename,
hypothesisEquality,
setEquality,
applyEquality,
hypothesis,
sqequalHypSubstitution,
sqequalRule,
universeEquality,
lemma_by_obid,
isectElimination,
because_Cache,
independent_isectElimination,
independent_functionElimination,
functionEquality,
cumulativity
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} Type]. \mforall{}[Q:T {}\mrightarrow{} \mBbbP{}]. (EquivRel(T;x,y.R[x;y]) {}\mRightarrow{} EquivRel(\{z:T| Q[z]\} ;x,y.R\000C[x;y]))
Date html generated:
2016_05_13-PM-04_15_04
Last ObjectModification:
2015_12_26-AM-11_30_03
Theory : rel_1
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