Nuprl Lemma : uconnex_functionality_wrt_implies
∀[T:Type]. ∀[R,R':T ⟶ T ⟶ ℙ].
((∀[x,y:T]. {R[x;y] ⇒ R'[x;y]}) ⇒ {uconnex(T; x,y.R[x;y]) ⇒ uconnex(T; x,y.R'[x;y])})
Proof
Definitions occuring in Statement :
uconnex: uconnex(T; x,y.R[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s1;s2],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uconnex: uconnex(T; x,y.R[x; y]),
guard: {T},
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s],
or: P ∨ Q
Lemmas referenced :
uall_wf,
or_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesis,
functionEquality,
universeEquality,
unionElimination,
independent_functionElimination,
inlFormation,
inrFormation
Latex:
\mforall{}[T:Type]. \mforall{}[R,R':T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
((\mforall{}[x,y:T]. \{R[x;y] {}\mRightarrow{} R'[x;y]\}) {}\mRightarrow{} \{uconnex(T; x,y.R[x;y]) {}\mRightarrow{} uconnex(T; x,y.R'[x;y])\})
Date html generated:
2016_10_21-AM-09_42_25
Last ObjectModification:
2016_08_01-PM-09_49_09
Theory : rel_1
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