Nuprl Lemma : homeo-image-boundary
∀[X,Y:Type]. ∀[dX:metric(X)]. ∀[dY:metric(Y)].
∀h:homeomorphic(X;dX;Y;dY)
∀[A:Type]
homeo-image(m-boundary(X;dX;A);Y;dY;h) ≡ m-boundary(Y;dY;homeo-image(A;Y;dY;h)) supposing metric-subspace(X;dX;A)
supposing PtwiseCONT(fst(h):X ⟶ Y) ∧ PtwiseCONT(snd(h):Y ⟶ X)
Proof
Definitions occuring in Statement :
homeo-image: homeo-image(A;Y;dY;h),
m-ptwise-cont: PtwiseCONT(f:X ⟶ Y),
m-boundary: m-boundary(X;d;A),
metric-subspace: metric-subspace(X;d;A),
homeomorphic: homeomorphic(X;dX;Y;dY),
metric: metric(X),
ext-eq: A ≡ B,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
and: P ∧ Q,
metric-subspace: metric-subspace(X;d;A),
uiff: uiff(P;Q),
homeomorphic: homeomorphic(X;dX;Y;dY),
exists: ∃x:A. B[x],
sq_exists: ∃x:A [B[x]],
ext-eq: A ≡ B,
subtype_rel: A ⊆r B,
homeo-image: homeo-image(A;Y;dY;h),
pi1: fst(t),
m-boundary: m-boundary(X;d;A),
mfun: FUN(X ⟶ Y),
prop: ℙ,
not: ¬A,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
false: False,
guard: {T},
pi2: snd(t),
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
rneq: x ≠ y,
nat_plus: ℕ+,
m-interior-point: m-interior-point(X;d;A;p),
respects-equality: respects-equality(S;T),
rev_uimplies: rev_uimplies(P;Q),
m-ptwise-cont: PtwiseCONT(f:X ⟶ Y),
rless: x < y,
squash: ↓T,
is-mfun: f:FUN(X;Y),
rge: x ≥ y,
rgt: x > y
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(m-boundary(X;dX;A);Y;dY;h) \mequiv{} m-boundary(Y;dY;homeo-image(A;Y;dY;h))
supposing metric-subspace(X;dX;A)
supposing PtwiseCONT(fst(h):X {}\mrightarrow{} Y) \mwedge{} PtwiseCONT(snd(h):Y {}\mrightarrow{} X)
Date html generated:
2020_05_20-AM-11_52_37
Last ObjectModification:
2020_01_06-PM-00_16_47
Theory : reals
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