Theorem cfilufg 24231

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 24227	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) → 𝐹 ∈ (fBas‘𝑋))
2		fgcl 23816	. . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
3		filfbas 23786	. . 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 23810	. . . . . . 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 3959	. . . . 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 5686	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑎 × 𝑎) = (𝑏 × 𝑏))
12	11	sseq1d 3990	. . . . . 6 ⊢ (𝑎 = 𝑏 → ((𝑎 × 𝑎) ⊆ 𝑣 ↔ (𝑏 × 𝑏) ⊆ 𝑣))
13	12	rspcev 3601	. . . . 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 24226	. . . . . 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 3234	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) ∧ 𝑣 ∈ 𝑈) → ∃𝑏 ∈ 𝐹 (𝑏 × 𝑏) ⊆ 𝑣)
18	14, 17	r19.29a 3148	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) ∧ 𝑣 ∈ 𝑈) → ∃𝑎 ∈ (𝑋filGen𝐹)(𝑎 × 𝑎) ⊆ 𝑣)
19	18	ralrimiva 3132	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) → ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ (𝑋filGen𝐹)(𝑎 × 𝑎) ⊆ 𝑣)
20		iscfilu 24226	. . 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 2108  ∀wral 3051  ∃wrex 3060   ⊆ wss 3926   × cxp 5652  ‘cfv 6531  (class class class)co 7405  fBascfbas 21303  filGencfg 21304  Filcfil 23783  UnifOncust 24138  CauFil_uccfilu 24224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-fbas 21312  df-fg 21313  df-fil 23784  df-ust 24139  df-cfilu 24225

This theorem is referenced by:  ucnextcn  24242