Theorem iscusp2 24189

Description: The predicate "𝑊 is a complete uniform space." (Contributed by Thierry Arnoux, 15-Dec-2017.)

Hypotheses

Ref	Expression
iscusp2.1	⊢ 𝐵 = (Base‘𝑊)
iscusp2.2	⊢ 𝑈 = (UnifSt‘𝑊)
iscusp2.3	⊢ 𝐽 = (TopOpen‘𝑊)

Assertion

Ref	Expression
iscusp2	⊢ (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘𝐵)(𝑐 ∈ (CauFilu‘𝑈) → (𝐽 fLim 𝑐) ≠ ∅)))

Distinct variable group:   𝑊,𝑐

Allowed substitution hints:   𝐵(𝑐)   𝑈(𝑐)   𝐽(𝑐)

Proof of Theorem iscusp2

Step	Hyp	Ref	Expression
1		iscusp 24186	. 2 ⊢ (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))
2		iscusp2.1	. . . . 5 ⊢ 𝐵 = (Base‘𝑊)
3	2	fveq2i 6861	. . . 4 ⊢ (Fil‘𝐵) = (Fil‘(Base‘𝑊))
4		iscusp2.2	. . . . . . 7 ⊢ 𝑈 = (UnifSt‘𝑊)
5	4	fveq2i 6861	. . . . . 6 ⊢ (CauFilu‘𝑈) = (CauFilu‘(UnifSt‘𝑊))
6	5	eleq2i 2820	. . . . 5 ⊢ (𝑐 ∈ (CauFilu‘𝑈) ↔ 𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)))
7		iscusp2.3	. . . . . . 7 ⊢ 𝐽 = (TopOpen‘𝑊)
8	7	oveq1i 7397	. . . . . 6 ⊢ (𝐽 fLim 𝑐) = ((TopOpen‘𝑊) fLim 𝑐)
9	8	neeq1i 2989	. . . . 5 ⊢ ((𝐽 fLim 𝑐) ≠ ∅ ↔ ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)
10	6, 9	imbi12i 350	. . . 4 ⊢ ((𝑐 ∈ (CauFilu‘𝑈) → (𝐽 fLim 𝑐) ≠ ∅) ↔ (𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))
11	3, 10	raleqbii 3317	. . 3 ⊢ (∀𝑐 ∈ (Fil‘𝐵)(𝑐 ∈ (CauFilu‘𝑈) → (𝐽 fLim 𝑐) ≠ ∅) ↔ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))
12	11	anbi2i 623	. 2 ⊢ ((𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘𝐵)(𝑐 ∈ (CauFilu‘𝑈) → (𝐽 fLim 𝑐) ≠ ∅)) ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))
13	1, 12	bitr4i 278	1 ⊢ (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘𝐵)(𝑐 ∈ (CauFilu‘𝑈) → (𝐽 fLim 𝑐) ≠ ∅)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∅c0 4296  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  TopOpenctopn 17384  Filcfil 23732   fLim cflim 23821  UnifStcuss 24141  UnifSpcusp 24142  CauFil_uccfilu 24173  CUnifSpccusp 24184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-ov 7390  df-cusp 24185

This theorem is referenced by:  cmetcusp1  25253