Nuprl Lemma : meqfun-equiv-rel-mfun
∀[A,X:Type]. ∀[dA:metric(A)]. ∀[d:metric(X)]. EquivRel(FUN(A ⟶ X);f,g.meqfun(d;A;f;g))
Proof
Definitions occuring in Statement :
meqfun: meqfun(d;A;f;g),
mfun: FUN(X ⟶ Y),
metric: metric(X),
equiv_rel: EquivRel(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
equiv_rel: EquivRel(T;x,y.E[x; y]),
and: P ∧ Q,
refl: Refl(T;x,y.E[x; y]),
all: ∀x:A. B[x],
meqfun: meqfun(d;A;f;g),
mfun: FUN(X ⟶ Y),
cand: A c∧ B,
sym: Sym(T;x,y.E[x; y]),
implies: P ⇒ Q,
prop: ℙ,
trans: Trans(T;x,y.E[x; y]),
meq: x ≡ y,
metric: metric(X),
guard: {T},
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
meq-same,
mfun_wf,
meqfun_wf,
req_witness,
int-to-real_wf,
metric_wf,
istype-universe,
meq_functionality,
meq_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
setElimination,
rename,
hypothesis,
universeIsType,
because_Cache,
sqequalRule,
productElimination,
independent_pairEquality,
lambdaEquality_alt,
dependent_functionElimination,
natural_numberEquality,
independent_functionElimination,
functionIsTypeImplies,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
instantiate,
universeEquality,
independent_isectElimination
Latex:
\mforall{}[A,X:Type]. \mforall{}[dA:metric(A)]. \mforall{}[d:metric(X)]. EquivRel(FUN(A {}\mrightarrow{} X);f,g.meqfun(d;A;f;g))
Date html generated:
2019_10_30-AM-06_29_46
Last ObjectModification:
2019_10_02-AM-10_04_49
Theory : reals
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