Nuprl Lemma : uequiv_rel_functionality_wrt_iff
∀[T,T':Type]. ∀[E:T ⟶ T ⟶ ℙ]. ∀[E':T' ⟶ T' ⟶ ℙ].
(∀[x,y:T]. (E[x;y] ⇐⇒ E'[x;y])) ⇒ (UniformEquivRel(T;x,y.E[x;y]) ⇐⇒ UniformEquivRel(T';x,y.E'[x;y]))
supposing T = T' ∈ Type
Proof
Definitions occuring in Statement :
uequiv_rel: UniformEquivRel(T;x,y.E[x; y]),
uimplies: b supposing a,
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,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
uequiv_rel: UniformEquivRel(T;x,y.E[x; y]),
urefl: UniformlyRefl(T;x,y.E[x; y]),
so_apply: x[s1;s2],
usym: UniformlySym(T;x,y.E[x; y]),
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
utrans: UniformlyTrans(T;x,y.E[x; y]),
so_lambda: λ2x y.t[x; y],
so_lambda: λ2x.t[x],
guard: {T},
so_apply: x[s]
Lemmas referenced :
subtype_rel_self,
uequiv_rel_wf,
uall_wf,
iff_wf,
subtype_rel_weakening,
ext-eq_weakening,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
axiomEquality,
hypothesis,
thin,
rename,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
promote_hyp,
isectElimination,
hyp_replacement,
hypothesisEquality,
equalitySymmetry,
because_Cache,
independent_functionElimination,
applyEquality,
sqequalRule,
instantiate,
extract_by_obid,
lambdaEquality,
independent_isectElimination,
universeEquality,
functionEquality,
cumulativity
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{} (UniformEquivRel(T;x,y.E[x;y]) \mLeftarrow{}{}\mRightarrow{} UniformEquivRel(T';x,y.E'[x;y]))
supposing T = T'
Date html generated:
2019_06_20-PM-00_29_04
Last ObjectModification:
2018_08_25-AM-08_25_07
Theory : rel_1
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