Nuprl Lemma : connex_functionality_wrt_iff
∀[T:Type]. ∀[R,R':T ⟶ T ⟶ ℙ]. ((∀x,y:T. (R[x;y] ⇐⇒ R'[x;y])) ⇒ (Connex(T;x,y.R[x;y]) ⇐⇒ Connex(T;x,y.R'[x;y])))
Proof
Definitions occuring in Statement :
connex: Connex(T;x,y.R[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
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 :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
connex: Connex(T;x,y.R[x; y]),
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
all: ∀x:A. B[x],
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
or: P ∨ Q,
guard: {T}
Lemmas referenced :
all_wf,
iff_wf,
or_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesis,
functionEquality,
universeEquality,
independent_pairFormation,
because_Cache,
addLevel,
productElimination,
impliesFunctionality,
allFunctionality,
orFunctionality,
dependent_functionElimination,
independent_functionElimination,
allLevelFunctionality,
orLevelFunctionality
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{} (Connex(T;x,y.R[x;y]) \mLeftarrow{}{}\mRightarrow{} Connex(T;x,y.R'[x;y])))
Date html generated:
2016_10_21-AM-09_42_17
Last ObjectModification:
2016_08_01-PM-09_49_21
Theory : rel_1
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