Nuprl Lemma : restriction-to-field
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀[P:T ⟶ ℙ]. ((∀x,y:T. ((R x y) ⇒ ((P x) ∧ (P y)))) ⇒ (∀x,y:T. (R|P x y ⇐⇒ R x y)))
Proof
Definitions occuring in Statement :
rel-restriction: R|P,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
rel-restriction: R|P,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T}
Lemmas referenced :
and_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
cut,
lemma_by_obid,
isectElimination,
applyEquality,
hypothesisEquality,
lambdaEquality,
functionEquality,
cumulativity,
universeEquality,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}].
((\mforall{}x,y:T. ((R x y) {}\mRightarrow{} ((P x) \mwedge{} (P y)))) {}\mRightarrow{} (\mforall{}x,y:T. (R|P x y \mLeftarrow{}{}\mRightarrow{} R x y)))
Date html generated:
2016_05_14-AM-06_06_08
Last ObjectModification:
2015_12_26-AM-11_32_35
Theory : relations
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