Theorem iscusp2 24327

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 24324	. 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 6910	. . . 4 ⊢ (Fil‘𝐵) = (Fil‘(Base‘𝑊))
4		iscusp2.2	. . . . . . 7 ⊢ 𝑈 = (UnifSt‘𝑊)
5	4	fveq2i 6910	. . . . . 6 ⊢ (CauFilu‘𝑈) = (CauFilu‘(UnifSt‘𝑊))
6	5	eleq2i 2831	. . . . 5 ⊢ (𝑐 ∈ (CauFilu‘𝑈) ↔ 𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)))
7		iscusp2.3	. . . . . . 7 ⊢ 𝐽 = (TopOpen‘𝑊)
8	7	oveq1i 7441	. . . . . 6 ⊢ (𝐽 fLim 𝑐) = ((TopOpen‘𝑊) fLim 𝑐)
9	8	neeq1i 3003	. . . . 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 3342	. . 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 1537   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  ∅c0 4339  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  TopOpenctopn 17468  Filcfil 23869   fLim cflim 23958  UnifStcuss 24278  UnifSpcusp 24279  CauFil_uccfilu 24311  CUnifSpccusp 24322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-cusp 24323

This theorem is referenced by:  cmetcusp1  25401