Nuprl Lemma : equiv_rel_functionality_wrt_iff
∀[T,T':Type]. ∀[E:T ⟶ T ⟶ ℙ]. ∀[E':T' ⟶ T' ⟶ ℙ].
(∀x,y:T. (E[x;y] ⇐⇒ E'[x;y])) ⇒ (EquivRel(T;x,y.E[x;y]) ⇐⇒ EquivRel(T';x,y.E'[x;y])) supposing T = T' ∈ Type
Proof
Definitions occuring in Statement :
equiv_rel: EquivRel(T;x,y.E[x; y]),
uimplies: b supposing a,
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,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
guard: {T},
so_apply: x[s],
refl: Refl(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
equiv_rel: EquivRel(T;x,y.E[x; y]),
iff: P ⇐⇒ Q,
and: P ∧ Q,
all: ∀x:A. B[x],
rev_implies: P ⇐ Q
Lemmas referenced :
equal_wf,
ext-eq_weakening,
subtype_rel_weakening,
iff_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
axiomEquality,
hypothesis,
thin,
rename,
lambdaFormation,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
independent_isectElimination,
because_Cache,
instantiate,
universeEquality,
functionEquality,
cumulativity,
productElimination,
productEquality,
addLevel,
independent_pairFormation,
impliesFunctionality,
allFunctionality,
dependent_functionElimination,
independent_functionElimination,
andLevelFunctionality,
allLevelFunctionality,
impliesLevelFunctionality,
equalitySymmetry
Latex:
\mforall{}[T,T':Type]. \mforall{}[E:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. \mforall{}[E':T' {}\mrightarrow{} T' {}\mrightarrow{} \mBbbP{}].
(\mforall{}x,y:T. (E[x;y] \mLeftarrow{}{}\mRightarrow{} E'[x;y])) {}\mRightarrow{} (EquivRel(T;x,y.E[x;y]) \mLeftarrow{}{}\mRightarrow{} EquivRel(T';x,y.E'[x;y]))
supposing T = T'
Date html generated:
2016_05_13-PM-04_15_23
Last ObjectModification:
2016_01_05-PM-01_47_39
Theory : rel_1
Home
Index