Theorem cfilufg 23455

Description: The filter generated by a Cauchy filter base is still a Cauchy filter base. (Contributed by Thierry Arnoux, 24-Jan-2018.)

Assertion

Ref	Expression
cfilufg	⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) → (𝑋filGen𝐹) ∈ (CauFilu‘𝑈))

Proof of Theorem cfilufg

Dummy variables 𝑎 𝑏 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cfilufbas 23451	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) → 𝐹 ∈ (fBas‘𝑋))
2		fgcl 23039	. . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
3		filfbas 23009	. . 3 ⊢ ((𝑋filGen𝐹) ∈ (Fil‘𝑋) → (𝑋filGen𝐹) ∈ (fBas‘𝑋))
4	1, 2, 3	3syl 18	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) → (𝑋filGen𝐹) ∈ (fBas‘𝑋))
5	1	ad3antrrr 727	. . . . . . 7 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) ∧ 𝑣 ∈ 𝑈) ∧ 𝑏 ∈ 𝐹) ∧ (𝑏 × 𝑏) ⊆ 𝑣) → 𝐹 ∈ (fBas‘𝑋))
6		ssfg 23033	. . . . . . 7 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
7	5, 6	syl 17	. . . . . 6 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) ∧ 𝑣 ∈ 𝑈) ∧ 𝑏 ∈ 𝐹) ∧ (𝑏 × 𝑏) ⊆ 𝑣) → 𝐹 ⊆ (𝑋filGen𝐹))
8		simplr 766	. . . . . 6 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) ∧ 𝑣 ∈ 𝑈) ∧ 𝑏 ∈ 𝐹) ∧ (𝑏 × 𝑏) ⊆ 𝑣) → 𝑏 ∈ 𝐹)
9	7, 8	sseldd 3921	. . . . 5 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) ∧ 𝑣 ∈ 𝑈) ∧ 𝑏 ∈ 𝐹) ∧ (𝑏 × 𝑏) ⊆ 𝑣) → 𝑏 ∈ (𝑋filGen𝐹))
10		id 22	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → 𝑎 = 𝑏)
11	10	sqxpeqd 5616	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑎 × 𝑎) = (𝑏 × 𝑏))
12	11	sseq1d 3951	. . . . . 6 ⊢ (𝑎 = 𝑏 → ((𝑎 × 𝑎) ⊆ 𝑣 ↔ (𝑏 × 𝑏) ⊆ 𝑣))
13	12	rspcev 3559	. . . . 5 ⊢ ((𝑏 ∈ (𝑋filGen𝐹) ∧ (𝑏 × 𝑏) ⊆ 𝑣) → ∃𝑎 ∈ (𝑋filGen𝐹)(𝑎 × 𝑎) ⊆ 𝑣)
14	9, 13	sylancom 588	. . . 4 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) ∧ 𝑣 ∈ 𝑈) ∧ 𝑏 ∈ 𝐹) ∧ (𝑏 × 𝑏) ⊆ 𝑣) → ∃𝑎 ∈ (𝑋filGen𝐹)(𝑎 × 𝑎) ⊆ 𝑣)
15		iscfilu 23450	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu‘𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑏 ∈ 𝐹 (𝑏 × 𝑏) ⊆ 𝑣)))
16	15	simplbda 500	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) → ∀𝑣 ∈ 𝑈 ∃𝑏 ∈ 𝐹 (𝑏 × 𝑏) ⊆ 𝑣)
17	16	r19.21bi 3133	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) ∧ 𝑣 ∈ 𝑈) → ∃𝑏 ∈ 𝐹 (𝑏 × 𝑏) ⊆ 𝑣)
18	14, 17	r19.29a 3216	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) ∧ 𝑣 ∈ 𝑈) → ∃𝑎 ∈ (𝑋filGen𝐹)(𝑎 × 𝑎) ⊆ 𝑣)
19	18	ralrimiva 3108	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) → ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ (𝑋filGen𝐹)(𝑎 × 𝑎) ⊆ 𝑣)
20		iscfilu 23450	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ((𝑋filGen𝐹) ∈ (CauFilu‘𝑈) ↔ ((𝑋filGen𝐹) ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ (𝑋filGen𝐹)(𝑎 × 𝑎) ⊆ 𝑣)))
21	20	adantr 481	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) → ((𝑋filGen𝐹) ∈ (CauFilu‘𝑈) ↔ ((𝑋filGen𝐹) ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ (𝑋filGen𝐹)(𝑎 × 𝑎) ⊆ 𝑣)))
22	4, 19, 21	mpbir2and 710	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) → (𝑋filGen𝐹) ∈ (CauFilu‘𝑈))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ⊆ wss 3886   × cxp 5582  ‘cfv 6426  (class class class)co 7267  fBascfbas 20595  filGencfg 20596  Filcfil 23006  UnifOncust 23361  CauFil_uccfilu 23448

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 5221  ax-nul 5228  ax-pow 5286  ax-pr 5350  ax-un 7578

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-nel 3050  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3431  df-sbc 3716  df-csb 3832  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5074  df-opab 5136  df-mpt 5157  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-res 5596  df-ima 5597  df-iota 6384  df-fun 6428  df-fn 6429  df-fv 6434  df-ov 7270  df-oprab 7271  df-mpo 7272  df-fbas 20604  df-fg 20605  df-fil 23007  df-ust 23362  df-cfilu 23449

This theorem is referenced by:  ucnextcn  23466