Nuprl Lemma : equiv_rel_self_functionality
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. (EquivRel(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 :
equiv_rel: EquivRel(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s1;s2],
all: ∀x:A. B[x],
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,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
so_apply: x[s1;s2],
rev_implies: P ⇐ Q,
so_lambda: λ2x y.t[x; y],
equiv_rel: EquivRel(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y])
Lemmas referenced :
equiv_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
lambdaFormation,
independent_pairFormation,
applyEquality,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
lambdaEquality,
hypothesis,
Error :functionIsType,
Error :universeIsType,
Error :inhabitedIsType,
universeEquality,
productElimination,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(EquivRel(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_06
Last ObjectModification:
2018_09_26-AM-11_56_05
Theory : rel_1
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