Nuprl Lemma : quotient-function-subtype
∀[X:Type]
∀[A:Type]. ∀[E:A ⟶ A ⟶ ℙ].
(EquivRel(A;a,b.E[a;b]) ⇒ ((X ⟶ (a,b:A//E[a;b])) ⊆r (f,g:X ⟶ A//fun-equiv(X;a,b.↓E[a;b];f;g))))
supposing X ⊆r Base
Proof
Definitions occuring in Statement :
fun-equiv: fun-equiv(X;a,b.E[a; b];f;g),
equiv_rel: EquivRel(T;x,y.E[x; y]),
quotient: x,y:A//B[x; y],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
squash: ↓T,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
base: Base,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
so_apply: x[s1;s2;s3],
fun-equiv: fun-equiv(X;a,b.E[a; b];f;g),
prop: ℙ,
so_lambda: λ2x y.t[x; y],
subtype_rel: A ⊆r B
Lemmas referenced :
quotient-dep-function-subtype,
equiv_rel_wf,
subtype_rel_wf,
base_wf
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
introduction,
independent_isectElimination,
lambdaFormation,
sqequalRule,
lambdaEquality,
applyEquality,
independent_functionElimination,
dependent_functionElimination,
axiomEquality,
functionEquality,
cumulativity,
universeEquality,
isect_memberEquality,
because_Cache
Latex:
\mforall{}[X:Type]
\mforall{}[A:Type]. \mforall{}[E:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}].
(EquivRel(A;a,b.E[a;b])
{}\mRightarrow{} ((X {}\mrightarrow{} (a,b:A//E[a;b])) \msubseteq{}r (f,g:X {}\mrightarrow{} A//fun-equiv(X;a,b.\mdownarrow{}E[a;b];f;g))))
supposing X \msubseteq{}r Base
Date html generated:
2016_05_14-AM-06_09_11
Last ObjectModification:
2015_12_26-AM-11_48_20
Theory : quot_1
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