Nuprl Lemma : xxtrans_functionality_wrt_breqv
∀[T:Type]. ∀[R,R':T ⟶ T ⟶ ℙ]. ((R <≡>{T} R') ⇒ (trans(T;R) ⇐⇒ trans(T;R')))
Proof
Definitions occuring in Statement :
xxtrans: trans(T;E),
binrel_eqv: E <≡>{T} E',
uall: ∀[x:A]. B[x],
prop: ℙ,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
xxtrans: trans(T;E),
trans: Trans(T;x,y.E[x; y]),
all: ∀x:A. B[x],
member: t ∈ T,
binrel_eqv: E <≡>{T} E',
prop: ℙ,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
guard: {T}
Lemmas referenced :
xxtrans_wf,
binrel_eqv_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
cut,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
productElimination,
applyEquality,
lambdaEquality,
sqequalRule,
universeEquality,
because_Cache,
lemma_by_obid,
isectElimination,
functionEquality,
cumulativity
Latex:
\mforall{}[T:Type]. \mforall{}[R,R':T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. ((R <\mequiv{}>\{T\} R') {}\mRightarrow{} (trans(T;R) \mLeftarrow{}{}\mRightarrow{} trans(T;R')))
Date html generated:
2016_05_15-PM-00_00_56
Last ObjectModification:
2015_12_26-PM-11_26_42
Theory : gen_algebra_1
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