Theorem cfiluweak 24488

Description: A Cauchy filter base is also a Cauchy filter base on any coarser uniform structure. (Contributed by Thierry Arnoux, 24-Jan-2018.)

Assertion

Ref	Expression
cfiluweak	⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝐹 ∈ (CauFilu‘𝑈))

Proof of Theorem cfiluweak

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

Step	Hyp	Ref	Expression
1		trust 24423	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
2		iscfilu 24481	. . . . . 6 ⊢ ((𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴) → (𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴))) ↔ (𝐹 ∈ (fBas‘𝐴) ∧ ∀𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑢)))
3	2	biimpa 482	. . . . 5 ⊢ (((𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴) ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) → (𝐹 ∈ (fBas‘𝐴) ∧ ∀𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑢))
4	1, 3	stoic3 1809	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) → (𝐹 ∈ (fBas‘𝐴) ∧ ∀𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑢))
5	4	simpld 500	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝐹 ∈ (fBas‘𝐴))
6		fbsspw 24026	. . . . 5 ⊢ (𝐹 ∈ (fBas‘𝐴) → 𝐹 ⊆ 𝒫 𝐴)
7	5, 6	syl 18	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝐹 ⊆ 𝒫 𝐴)
8		simp2 1155	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝐴 ⊆ 𝑋)
9	8	sspwd 4580	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝒫 𝐴 ⊆ 𝒫 𝑋)
10	7, 9	sstrd 3950	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝐹 ⊆ 𝒫 𝑋)
11		simp1 1154	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝑈 ∈ (UnifOn‘𝑋))
12	11	elfvexd 6924	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝑋 ∈ V)
13		fbasweak 24059	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝐴) ∧ 𝐹 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ V) → 𝐹 ∈ (fBas‘𝑋))
14	5, 10, 12, 13	syl3anc 1398	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝐹 ∈ (fBas‘𝑋))
15		sseq2 3966	. . . . . 6 ⊢ (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → ((𝑎 × 𝑎) ⊆ 𝑢 ↔ (𝑎 × 𝑎) ⊆ (𝑣 ∩ (𝐴 × 𝐴))))
16	15	rexbidv 3192	. . . . 5 ⊢ (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → (∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑢 ↔ ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ (𝑣 ∩ (𝐴 × 𝐴))))
17	4	simprd 501	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) → ∀𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑢)
18	17	adantr 486	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑣 ∈ 𝑈) → ∀𝑢 ∈ (𝑈 ↾t (𝐴 × 𝐴))∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑢)
19	11	adantr 486	. . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑣 ∈ 𝑈) → 𝑈 ∈ (UnifOn‘𝑋))
20	12	adantr 486	. . . . . . . 8 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑣 ∈ 𝑈) → 𝑋 ∈ V)
21	8	adantr 486	. . . . . . . 8 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑣 ∈ 𝑈) → 𝐴 ⊆ 𝑋)
22	20, 21	ssexd 5300	. . . . . . 7 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑣 ∈ 𝑈) → 𝐴 ∈ V)
23	22, 22	xpexd 7759	. . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑣 ∈ 𝑈) → (𝐴 × 𝐴) ∈ V)
24		simpr 490	. . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑣 ∈ 𝑈) → 𝑣 ∈ 𝑈)
25		elrestr 17506	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V ∧ 𝑣 ∈ 𝑈) → (𝑣 ∩ (𝐴 × 𝐴)) ∈ (𝑈 ↾t (𝐴 × 𝐴)))
26	19, 23, 24, 25	syl3anc 1398	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑣 ∈ 𝑈) → (𝑣 ∩ (𝐴 × 𝐴)) ∈ (𝑈 ↾t (𝐴 × 𝐴)))
27	16, 18, 26	rspcdva 3585	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑣 ∈ 𝑈) → ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ (𝑣 ∩ (𝐴 × 𝐴)))
28		inss1 4192	. . . . . 6 ⊢ (𝑣 ∩ (𝐴 × 𝐴)) ⊆ 𝑣
29		sstr 3948	. . . . . 6 ⊢ (((𝑎 × 𝑎) ⊆ (𝑣 ∩ (𝐴 × 𝐴)) ∧ (𝑣 ∩ (𝐴 × 𝐴)) ⊆ 𝑣) → (𝑎 × 𝑎) ⊆ 𝑣)
30	28, 29	mpan2 704	. . . . 5 ⊢ ((𝑎 × 𝑎) ⊆ (𝑣 ∩ (𝐴 × 𝐴)) → (𝑎 × 𝑎) ⊆ 𝑣)
31	30	reximi 3106	. . . 4 ⊢ (∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ (𝑣 ∩ (𝐴 × 𝐴)) → ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
32	27, 31	syl 18	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑣 ∈ 𝑈) → ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
33	32	ralrimiva 3160	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) → ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
34		iscfilu 24481	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu‘𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
35	34	3ad2ant1 1151	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) → (𝐹 ∈ (CauFilu‘𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
36	14, 33, 35	mpbir2and 726	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝐹 ∈ (CauFilu‘𝑈))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3082  ∃wrex 3092  Vcvv 3458   ∩ cin 3907   ⊆ wss 3908  𝒫 cpw 4567   × cxp 5664  ‘cfv 6543  (class class class)co 7423   ↾_t crest 17498  fBascfbas 21547  UnifOncust 24394  CauFil_uccfilu 24479

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-1st 7995  df-2nd 7996  df-rest 17500  df-fbas 21556  df-ust 24395  df-cfilu 24480

This theorem is used by: (None)