Theorem iscfilu 21902

Description: The predicate "𝐹 is a Cauchy filter base on uniform space 𝑈." (Contributed by Thierry Arnoux, 18-Nov-2017.)

Assertion

Ref	Expression
iscfilu	⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu‘𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))

Distinct variable groups:   𝑣,𝑎,𝐹   𝑣,𝑈

Allowed substitution hints:   𝑈(𝑎)   𝑋(𝑣,𝑎)

Proof of Theorem iscfilu

Dummy variables 𝑓 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elrnust 21838	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran UnifOn)
2		unieq 4380	. . . . . . . . 9 ⊢ (𝑢 = 𝑈 → ∪ 𝑢 = ∪ 𝑈)
3	2	dmeqd 5248	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → dom ∪ 𝑢 = dom ∪ 𝑈)
4	3	fveq2d 6107	. . . . . . 7 ⊢ (𝑢 = 𝑈 → (fBas‘dom ∪ 𝑢) = (fBas‘dom ∪ 𝑈))
5		raleq 3115	. . . . . . 7 ⊢ (𝑢 = 𝑈 → (∀𝑣 ∈ 𝑢 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣))
6	4, 5	rabeqbidv 3168	. . . . . 6 ⊢ (𝑢 = 𝑈 → {𝑓 ∈ (fBas‘dom ∪ 𝑢) ∣ ∀𝑣 ∈ 𝑢 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣} = {𝑓 ∈ (fBas‘dom ∪ 𝑈) ∣ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣})
7		df-cfilu 21901	. . . . . 6 ⊢ CauFilu = (𝑢 ∈ ∪ ran UnifOn ↦ {𝑓 ∈ (fBas‘dom ∪ 𝑢) ∣ ∀𝑣 ∈ 𝑢 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣})
8		fvex 6113	. . . . . . 7 ⊢ (fBas‘dom ∪ 𝑈) ∈ V
9	8	rabex 4740	. . . . . 6 ⊢ {𝑓 ∈ (fBas‘dom ∪ 𝑈) ∣ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣} ∈ V
10	6, 7, 9	fvmpt 6191	. . . . 5 ⊢ (𝑈 ∈ ∪ ran UnifOn → (CauFilu‘𝑈) = {𝑓 ∈ (fBas‘dom ∪ 𝑈) ∣ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣})
11	1, 10	syl 17	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (CauFilu‘𝑈) = {𝑓 ∈ (fBas‘dom ∪ 𝑈) ∣ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣})
12	11	eleq2d 2673	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu‘𝑈) ↔ 𝐹 ∈ {𝑓 ∈ (fBas‘dom ∪ 𝑈) ∣ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣}))
13		rexeq 3116	. . . . 5 ⊢ (𝑓 = 𝐹 → (∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
14	13	ralbidv 2969	. . . 4 ⊢ (𝑓 = 𝐹 → (∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
15	14	elrab 3331	. . 3 ⊢ (𝐹 ∈ {𝑓 ∈ (fBas‘dom ∪ 𝑈) ∣ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣} ↔ (𝐹 ∈ (fBas‘dom ∪ 𝑈) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
16	12, 15	syl6bb 275	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu‘𝑈) ↔ (𝐹 ∈ (fBas‘dom ∪ 𝑈) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
17		ustbas2 21839	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom ∪ 𝑈)
18	17	fveq2d 6107	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (fBas‘𝑋) = (fBas‘dom ∪ 𝑈))
19	18	eleq2d 2673	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (fBas‘𝑋) ↔ 𝐹 ∈ (fBas‘dom ∪ 𝑈)))
20	19	anbi1d 737	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ((𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣) ↔ (𝐹 ∈ (fBas‘dom ∪ 𝑈) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
21	16, 20	bitr4d 270	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu‘𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900   ⊆ wss 3540  ∪ cuni 4372   × cxp 5036  dom cdm 5038  ran crn 5039  ‘cfv 5804  fBascfbas 19555  UnifOncust 21813  CauFil_uccfilu 21900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-ust 21814  df-cfilu 21901

This theorem is referenced by:  cfilufbas  21903  cfiluexsm  21904  fmucnd  21906  cfilufg  21907  trcfilu  21908  cfiluweak  21909  neipcfilu  21910  cfilucfil  22174