Theorem iscusp 24329

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 6925	. . 3 ⊢ (𝑤 = 𝑊 → (Fil‘(Base‘𝑤)) = (Fil‘(Base‘𝑊)))
2		2fveq3 6925	. . . . 5 ⊢ (𝑤 = 𝑊 → (CauFilu‘(UnifSt‘𝑤)) = (CauFilu‘(UnifSt‘𝑊)))
3	2	eleq2d 2830	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) ↔ 𝑐 ∈ (CauFilu‘(UnifSt‘𝑊))))
4		fveq2 6920	. . . . . 6 ⊢ (𝑤 = 𝑊 → (TopOpen‘𝑤) = (TopOpen‘𝑊))
5	4	oveq1d 7463	. . . . 5 ⊢ (𝑤 = 𝑊 → ((TopOpen‘𝑤) fLim 𝑐) = ((TopOpen‘𝑊) fLim 𝑐))
6	5	neeq1d 3006	. . . 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 3354	. 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 24328	. 2 ⊢ CUnifSp = {𝑤 ∈ UnifSp ∣ ∀𝑐 ∈ (Fil‘(Base‘𝑤))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅)}
10	8, 9	elrab2 3711	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 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∅c0 4352  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  TopOpenctopn 17481  Filcfil 23874   fLim cflim 23963  UnifStcuss 24283  UnifSpcusp 24284  CauFil_uccfilu 24316  CUnifSpccusp 24327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-ov 7451  df-cusp 24328

This theorem is referenced by:  cuspusp  24330  cuspcvg  24331  iscusp2  24332  cmetcusp  25407