Theorem cfilufg 24207

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 24203	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) → 𝐹 ∈ (fBas‘𝑋))
2		fgcl 23793	. . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
3		filfbas 23763	. . 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 730	. . . . . . 7 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) ∧ 𝑣 ∈ 𝑈) ∧ 𝑏 ∈ 𝐹) ∧ (𝑏 × 𝑏) ⊆ 𝑣) → 𝐹 ∈ (fBas‘𝑋))
6		ssfg 23787	. . . . . . 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 768	. . . . . 6 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) ∧ 𝑣 ∈ 𝑈) ∧ 𝑏 ∈ 𝐹) ∧ (𝑏 × 𝑏) ⊆ 𝑣) → 𝑏 ∈ 𝐹)
9	7, 8	sseldd 3930	. . . . 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 5646	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑎 × 𝑎) = (𝑏 × 𝑏))
12	11	sseq1d 3961	. . . . . 6 ⊢ (𝑎 = 𝑏 → ((𝑎 × 𝑎) ⊆ 𝑣 ↔ (𝑏 × 𝑏) ⊆ 𝑣))
13	12	rspcev 3572	. . . . 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 24202	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu‘𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑏 ∈ 𝐹 (𝑏 × 𝑏) ⊆ 𝑣)))
16	15	simplbda 499	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) → ∀𝑣 ∈ 𝑈 ∃𝑏 ∈ 𝐹 (𝑏 × 𝑏) ⊆ 𝑣)
17	16	r19.21bi 3224	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) ∧ 𝑣 ∈ 𝑈) → ∃𝑏 ∈ 𝐹 (𝑏 × 𝑏) ⊆ 𝑣)
18	14, 17	r19.29a 3140	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) ∧ 𝑣 ∈ 𝑈) → ∃𝑎 ∈ (𝑋filGen𝐹)(𝑎 × 𝑎) ⊆ 𝑣)
19	18	ralrimiva 3124	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) → ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ (𝑋filGen𝐹)(𝑎 × 𝑎) ⊆ 𝑣)
20		iscfilu 24202	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ((𝑋filGen𝐹) ∈ (CauFilu‘𝑈) ↔ ((𝑋filGen𝐹) ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ (𝑋filGen𝐹)(𝑎 × 𝑎) ⊆ 𝑣)))
21	20	adantr 480	. 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 713	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) → (𝑋filGen𝐹) ∈ (CauFilu‘𝑈))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056   ⊆ wss 3897   × cxp 5612  ‘cfv 6481  (class class class)co 7346  fBascfbas 21279  filGencfg 21280  Filcfil 23760  UnifOncust 24115  CauFil_uccfilu 24200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-fbas 21288  df-fg 21289  df-fil 23761  df-ust 24116  df-cfilu 24201

This theorem is referenced by:  ucnextcn  24218