Theorem iscfilu 24615

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		elfvunirn 6920	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran UnifOn)
2		unieq 4877	. . . . . . . . 9 ⊢ (𝑢 = 𝑈 → ∪ 𝑢 = ∪ 𝑈)
3	2	dmeqd 5888	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → dom ∪ 𝑢 = dom ∪ 𝑈)
4	3	fveq2d 6894	. . . . . . 7 ⊢ (𝑢 = 𝑈 → (fBas‘dom ∪ 𝑢) = (fBas‘dom ∪ 𝑈))
5		raleq 3316	. . . . . . 7 ⊢ (𝑢 = 𝑈 → (∀𝑣 ∈ 𝑢 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣))
6	4, 5	rabeqbidv 3429	. . . . . 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 24614	. . . . . 6 ⊢ CauFilu = (𝑢 ∈ ∪ ran UnifOn ↦ {𝑓 ∈ (fBas‘dom ∪ 𝑢) ∣ ∀𝑣 ∈ 𝑢 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣})
8		fvex 6903	. . . . . . 7 ⊢ (fBas‘dom ∪ 𝑈) ∈ V
9	8	rabex 5300	. . . . . 6 ⊢ {𝑓 ∈ (fBas‘dom ∪ 𝑈) ∣ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣} ∈ V
10	6, 7, 9	fvmpt 6998	. . . . 5 ⊢ (𝑈 ∈ ∪ ran UnifOn → (CauFilu‘𝑈) = {𝑓 ∈ (fBas‘dom ∪ 𝑈) ∣ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣})
11	1, 10	syl 18	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (CauFilu‘𝑈) = {𝑓 ∈ (fBas‘dom ∪ 𝑈) ∣ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣})
12	11	eleq2d 2846	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu‘𝑈) ↔ 𝐹 ∈ {𝑓 ∈ (fBas‘dom ∪ 𝑈) ∣ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣}))
13		rexeq 3315	. . . . 5 ⊢ (𝑓 = 𝐹 → (∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
14	13	ralbidv 3185	. . . 4 ⊢ (𝑓 = 𝐹 → (∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
15	14	elrab 3644	. . 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 290	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu‘𝑈) ↔ (𝐹 ∈ (fBas‘dom ∪ 𝑈) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
17		ustbas2 24553	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom ∪ 𝑈)
18	17	fveq2d 6894	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (fBas‘𝑋) = (fBas‘dom ∪ 𝑈))
19	18	eleq2d 2846	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (fBas‘𝑋) ↔ 𝐹 ∈ (fBas‘dom ∪ 𝑈)))
20	19	anbi1d 643	. 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 285	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu‘𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  {crab 3412   ⊆ wss 3896  ∪ cuni 4866   × cxp 5649  dom cdm 5651  ran crn 5652  ‘cfv 6543  fBascfbas 21675  UnifOncust 24528  CauFil_uccfilu 24613

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-iota 6497  df-fun 6545  df-fv 6551  df-ust 24529  df-cfilu 24614

This theorem is used by:  cfilufbas  24616  cfiluexsm  24617  fmucnd  24619  cfilufg  24620  trcfilu  24621  cfiluweak  24622  neipcfilu  24623  cfilucfil  24887