Nuprl Lemma : uconnex_functionality_wrt_iff
∀[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,
so_apply: x[s1;s2],
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,
uconnex: uconnex(T; x,y.R[x; y]),
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
or: P ∨ Q,
so_apply: x[s1;s2],
prop: ℙ,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q
Lemmas referenced :
uall_wf,
or_wf,
iff_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
unionElimination,
inlFormation,
productElimination,
independent_functionElimination,
applyEquality,
sqequalRule,
inrFormation,
introduction,
extract_by_obid,
lambdaEquality,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R,R':T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
((\mforall{}[x,y:T]. (R[x;y] \mLeftarrow{}{}\mRightarrow{} R'[x;y])) {}\mRightarrow{} (uconnex(T; x,y.R[x;y]) \mLeftarrow{}{}\mRightarrow{} uconnex(T; x,y.R'[x;y])))
Date html generated:
2019_06_20-PM-00_29_20
Last ObjectModification:
2018_08_25-AM-08_24_26
Theory : rel_1
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