Nuprl Lemma : uequiv_rel_subtyping
∀[T:Type]. ∀[R:T ⟶ T ⟶ Type]. ∀[Q:T ⟶ ℙ]. (UniformEquivRel(T;x,y.R[x;y]) ⇒ UniformEquivRel({z:T| Q[z]} ;x,y.R[x;y])\000C)
Proof
Definitions occuring in Statement :
uequiv_rel: UniformEquivRel(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 :
uequiv_rel: UniformEquivRel(T;x,y.E[x; y]),
utrans: UniformlyTrans(T;x,y.E[x; y]),
usym: UniformlySym(T;x,y.E[x; y]),
urefl: UniformlyRefl(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
so_apply: x[s1;s2],
guard: {T}
Lemmas referenced :
set_wf,
uall_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesis,
universeEquality,
independent_pairFormation,
setElimination,
rename,
because_Cache,
productEquality,
functionEquality,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} Type]. \mforall{}[Q:T {}\mrightarrow{} \mBbbP{}].
(UniformEquivRel(T;x,y.R[x;y]) {}\mRightarrow{} UniformEquivRel(\{z:T| Q[z]\} ;x,y.R[x;y]))
Date html generated:
2016_10_21-AM-09_41_58
Last ObjectModification:
2016_08_01-PM-09_49_24
Theory : rel_1
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