Nuprl Lemma : ss-fun-eq
∀[X,Y:SeparationSpace]. ∀[f,g:Point(X ⟶ Y)]. uiff(f ≡ g;∀a:Point(X). f(a) ≡ g(a))
Proof
Definitions occuring in Statement :
ss-ap: f(x),
ss-fun: X ⟶ Y,
ss-eq: x ≡ y,
ss-point: Point(ss),
separation-space: SeparationSpace,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
all: ∀x:A. B[x],
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
ss-ap: f(x),
top: Top,
member: t ∈ T,
ss-eq: x ≡ y,
uall: ∀[x:A]. B[x]
Lemmas referenced :
separation-space_wf,
ss-fun_wf,
all_wf,
exists_wf,
not_wf,
ss-point_wf,
ss-ap_wf,
ss-sep_wf,
ss-fun-sep
Rules used in proof :
productElimination,
because_Cache,
dependent_functionElimination,
lambdaEquality,
hypothesisEquality,
dependent_pairFormation,
independent_functionElimination,
lambdaFormation,
independent_pairFormation,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
sqequalRule,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[X,Y:SeparationSpace]. \mforall{}[f,g:Point(X {}\mrightarrow{} Y)]. uiff(f \mequiv{} g;\mforall{}a:Point(X). f(a) \mequiv{} g(a))
Date html generated:
2018_07_29-AM-10_11_49
Last ObjectModification:
2018_07_04-PM-00_06_56
Theory : constructive!algebra
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