Nuprl Lemma : ss-ap_functionality
∀[X,Y:SeparationSpace]. ∀[f,g:Point(X ⟶ Y)]. ∀[x,x':Point(X)]. (f(x) ≡ g(x')) supposing (x ≡ x' and f ≡ g)
Proof
Definitions occuring in Statement :
ss-ap: f(x),
ss-fun: X ⟶ Y,
ss-eq: x ≡ y,
ss-point: Point(ss),
separation-space: SeparationSpace,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
ss-function: ss-function(X;Y;f),
guard: {T},
squash: ↓T,
sq_stable: SqStable(P),
top: Top,
ss-ap: f(x),
prop: ℙ,
false: False,
implies: P ⇒ Q,
not: ¬A,
ss-eq: x ≡ y,
all: ∀x:A. B[x],
and: P ∧ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
ss-eq_transitivity,
sq_stable__ss-eq,
ss-fun-point,
separation-space_wf,
ss-point_wf,
ss-fun_wf,
ss-eq_wf,
ss-ap_wf,
ss-sep_wf,
ss-fun-eq
Rules used in proof :
imageElimination,
baseClosed,
imageMemberEquality,
independent_functionElimination,
applyEquality,
rename,
setElimination,
voidEquality,
voidElimination,
equalitySymmetry,
equalityTransitivity,
isect_memberEquality,
because_Cache,
lambdaEquality,
sqequalRule,
dependent_functionElimination,
independent_isectElimination,
productElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[X,Y:SeparationSpace]. \mforall{}[f,g:Point(X {}\mrightarrow{} Y)]. \mforall{}[x,x':Point(X)].
(f(x) \mequiv{} g(x')) supposing (x \mequiv{} x' and f \mequiv{} g)
Date html generated:
2018_07_29-AM-10_11_52
Last ObjectModification:
2018_07_04-PM-00_10_57
Theory : constructive!algebra
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