Theorem iscfilu 23440

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 23376	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran UnifOn)
2		unieq 4850	. . . . . . . . 9 ⊢ (𝑢 = 𝑈 → ∪ 𝑢 = ∪ 𝑈)
3	2	dmeqd 5814	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → dom ∪ 𝑢 = dom ∪ 𝑈)
4	3	fveq2d 6778	. . . . . . 7 ⊢ (𝑢 = 𝑈 → (fBas‘dom ∪ 𝑢) = (fBas‘dom ∪ 𝑈))
5		raleq 3342	. . . . . . 7 ⊢ (𝑢 = 𝑈 → (∀𝑣 ∈ 𝑢 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣))
6	4, 5	rabeqbidv 3420	. . . . . 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 23439	. . . . . 6 ⊢ CauFilu = (𝑢 ∈ ∪ ran UnifOn ↦ {𝑓 ∈ (fBas‘dom ∪ 𝑢) ∣ ∀𝑣 ∈ 𝑢 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣})
8		fvex 6787	. . . . . . 7 ⊢ (fBas‘dom ∪ 𝑈) ∈ V
9	8	rabex 5256	. . . . . 6 ⊢ {𝑓 ∈ (fBas‘dom ∪ 𝑈) ∣ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣} ∈ V
10	6, 7, 9	fvmpt 6875	. . . . 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 2824	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu‘𝑈) ↔ 𝐹 ∈ {𝑓 ∈ (fBas‘dom ∪ 𝑈) ∣ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣}))
13		rexeq 3343	. . . . 5 ⊢ (𝑓 = 𝐹 → (∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
14	13	ralbidv 3112	. . . 4 ⊢ (𝑓 = 𝐹 → (∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
15	14	elrab 3624	. . 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	bitrdi 287	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu‘𝑈) ↔ (𝐹 ∈ (fBas‘dom ∪ 𝑈) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
17		ustbas2 23377	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom ∪ 𝑈)
18	17	fveq2d 6778	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (fBas‘𝑋) = (fBas‘dom ∪ 𝑈))
19	18	eleq2d 2824	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (fBas‘𝑋) ↔ 𝐹 ∈ (fBas‘dom ∪ 𝑈)))
20	19	anbi1d 630	. 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 281	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu‘𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  {crab 3068   ⊆ wss 3887  ∪ cuni 4839   × cxp 5587  dom cdm 5589  ran crn 5590  ‘cfv 6433  fBascfbas 20585  UnifOncust 23351  CauFil_uccfilu 23438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441  df-ust 23352  df-cfilu 23439

This theorem is referenced by:  cfilufbas  23441  cfiluexsm  23442  fmucnd  23444  cfilufg  23445  trcfilu  23446  cfiluweak  23447  neipcfilu  23448  cfilucfil  23715