Nuprl Lemma : dep-fun-equiv-rel
∀[X:Type]. ∀[A:X ⟶ Type]. ∀[E:x:X ⟶ A[x] ⟶ A[x] ⟶ ℙ].
((∀x:X. EquivRel(A[x];a,b.E[x;a;b])) ⇒ EquivRel(x:X ⟶ A[x];f,g.dep-fun-equiv(X;x,a,b.E[x;a;b];f;g)))
Proof
Definitions occuring in Statement :
dep-fun-equiv: dep-fun-equiv(X;x,a,b.E[x; a; b];f;g),
equiv_rel: EquivRel(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2;s3],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
equiv_rel: EquivRel(T;x,y.E[x; y]),
and: P ∧ Q,
dep-fun-equiv: dep-fun-equiv(X;x,a,b.E[x; a; b];f;g),
refl: Refl(T;x,y.E[x; y]),
all: ∀x:A. B[x],
member: t ∈ T,
so_apply: x[s],
cand: A c∧ B,
sym: Sym(T;x,y.E[x; y]),
so_apply: x[s1;s2;s3],
subtype_rel: A ⊆r B,
prop: ℙ,
trans: Trans(T;x,y.E[x; y]),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
guard: {T}
Lemmas referenced :
subtype_rel_self,
equiv_rel_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
independent_pairFormation,
sqequalRule,
because_Cache,
Error :functionIsType,
Error :universeIsType,
applyEquality,
hypothesisEquality,
cut,
hypothesis,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
Error :lambdaEquality_alt,
Error :inhabitedIsType,
universeEquality,
dependent_functionElimination,
productElimination,
independent_functionElimination
Latex:
\mforall{}[X:Type]. \mforall{}[A:X {}\mrightarrow{} Type]. \mforall{}[E:x:X {}\mrightarrow{} A[x] {}\mrightarrow{} A[x] {}\mrightarrow{} \mBbbP{}].
((\mforall{}x:X. EquivRel(A[x];a,b.E[x;a;b]))
{}\mRightarrow{} EquivRel(x:X {}\mrightarrow{} A[x];f,g.dep-fun-equiv(X;x,a,b.E[x;a;b];f;g)))
Date html generated:
2019_06_20-PM-00_32_56
Last ObjectModification:
2019_03_19-AM-10_42_34
Theory : quot_1
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