Nuprl Lemma : homeo-image-inverse
∀[X,Y:Type]. ∀[dX:metric(X)]. ∀[dY:metric(Y)]. ∀[h:homeomorphic(X;dX;Y;dY)]. ∀[A:Type].
homeo-image(homeo-image(A;Y;dY;h);X;dX;homeo-inv(h)) ≡ A supposing metric-subspace(X;dX;A)
Proof
Definitions occuring in Statement :
homeo-image: homeo-image(A;Y;dY;h),
metric-subspace: metric-subspace(X;d;A),
homeo-inv: homeo-inv(h),
homeomorphic: homeomorphic(X;dX;Y;dY),
metric: metric(X),
ext-eq: A ≡ B,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
metric-subspace: metric-subspace(X;d;A),
and: P ∧ Q,
uiff: uiff(P;Q),
ext-eq: A ≡ B,
subtype_rel: A ⊆r B,
homeo-image: homeo-image(A;Y;dY;h),
prop: ℙ,
exists: ∃x:A. B[x],
homeomorphic: homeomorphic(X;dX;Y;dY),
homeo-inv: homeo-inv(h),
pi1: fst(t),
sq_exists: ∃x:A [B[x]],
mfun: FUN(X ⟶ Y),
is-mfun: f:FUN(X;Y),
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_apply: x[s],
guard: {T},
label: ...$L... t,
meq: x ≡ y,
req: x = y,
sq_stable: SqStable(P),
squash: ↓T,
rev_uimplies: rev_uimplies(P;Q)
Latex:
\mforall{}[X,Y:Type]. \mforall{}[dX:metric(X)]. \mforall{}[dY:metric(Y)]. \mforall{}[h:homeomorphic(X;dX;Y;dY)]. \mforall{}[A:Type].
homeo-image(homeo-image(A;Y;dY;h);X;dX;homeo-inv(h)) \mequiv{} A supposing metric-subspace(X;dX;A)
Date html generated:
2020_05_20-AM-11_51_10
Last ObjectModification:
2019_11_20-AM-10_57_17
Theory : reals
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