Nuprl Lemma : uequiv_rel_self_functionality
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
(UniformEquivRel(T;x,y.R[x;y]) ⇒ {∀[a,a',b,b':T]. (R[a;b] ⇒ R[a';b'] ⇒ (R[a;a'] ⇐⇒ R[b;b']))})
Proof
Definitions occuring in Statement :
uequiv_rel: UniformEquivRel(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s1;s2],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
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]),
member: t ∈ T,
prop: ℙ,
so_apply: x[s1;s2],
rev_implies: P ⇐ Q,
so_lambda: λ2x y.t[x; y]
Lemmas referenced :
uequiv_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
applyEquality,
hypothesisEquality,
Error :inhabitedIsType,
Error :universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
lambdaEquality,
hypothesis,
Error :functionIsType,
universeEquality,
independent_functionElimination,
because_Cache
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(UniformEquivRel(T;x,y.R[x;y])
{}\mRightarrow{} \{\mforall{}[a,a',b,b':T]. (R[a;b] {}\mRightarrow{} R[a';b'] {}\mRightarrow{} (R[a;a'] \mLeftarrow{}{}\mRightarrow{} R[b;b']))\})
Date html generated:
2019_06_20-PM-00_29_08
Last ObjectModification:
2018_09_26-AM-11_57_50
Theory : rel_1
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