Theorem iscusp 24222

Description: The predicate "𝑊 is a complete uniform space." (Contributed by Thierry Arnoux, 3-Dec-2017.)

Assertion

Ref	Expression
iscusp	⊢ (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))

Distinct variable group:   𝑊,𝑐

Proof of Theorem iscusp

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2fveq3 6877	. . 3 ⊢ (𝑤 = 𝑊 → (Fil‘(Base‘𝑤)) = (Fil‘(Base‘𝑊)))
2		2fveq3 6877	. . . . 5 ⊢ (𝑤 = 𝑊 → (CauFilu‘(UnifSt‘𝑤)) = (CauFilu‘(UnifSt‘𝑊)))
3	2	eleq2d 2819	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) ↔ 𝑐 ∈ (CauFilu‘(UnifSt‘𝑊))))
4		fveq2 6872	. . . . . 6 ⊢ (𝑤 = 𝑊 → (TopOpen‘𝑤) = (TopOpen‘𝑊))
5	4	oveq1d 7414	. . . . 5 ⊢ (𝑤 = 𝑊 → ((TopOpen‘𝑤) fLim 𝑐) = ((TopOpen‘𝑊) fLim 𝑐))
6	5	neeq1d 2990	. . . 4 ⊢ (𝑤 = 𝑊 → (((TopOpen‘𝑤) fLim 𝑐) ≠ ∅ ↔ ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))
7	3, 6	imbi12d 344	. . 3 ⊢ (𝑤 = 𝑊 → ((𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅) ↔ (𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))
8	1, 7	raleqbidv 3323	. 2 ⊢ (𝑤 = 𝑊 → (∀𝑐 ∈ (Fil‘(Base‘𝑤))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅) ↔ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))
9		df-cusp 24221	. 2 ⊢ CUnifSp = {𝑤 ∈ UnifSp ∣ ∀𝑐 ∈ (Fil‘(Base‘𝑤))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅)}
10	8, 9	elrab2 3672	1 ⊢ (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107   ≠ wne 2931  ∀wral 3050  ∅c0 4306  ‘cfv 6527  (class class class)co 7399  Basecbs 17213  TopOpenctopn 17420  Filcfil 23768   fLim cflim 23857  UnifStcuss 24177  UnifSpcusp 24178  CauFil_uccfilu 24209  CUnifSpccusp 24220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-ral 3051  df-rab 3414  df-v 3459  df-dif 3927  df-un 3929  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4881  df-br 5117  df-iota 6480  df-fv 6535  df-ov 7402  df-cusp 24221

This theorem is referenced by:  cuspusp  24223  cuspcvg  24224  iscusp2  24225  cmetcusp  25291