Theorem cfiluweak 24618

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 24553	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
2		iscfilu 24611	. . . . . 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 1808	. . . 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 24156	. . . . 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 4569	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝒫 𝐴 ⊆ 𝒫 𝑋)
10	7, 9	sstrd 3918	. . 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 6921	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝑋 ∈ V)
13		fbasweak 24189	. . 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 3934	. . . . . 6 ⊢ (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → ((𝑎 × 𝑎) ⊆ 𝑢 ↔ (𝑎 × 𝑎) ⊆ (𝑣 ∩ (𝐴 × 𝐴))))
16	15	rexbidv 3186	. . . . 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 5285	. . . . . . 7 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑣 ∈ 𝑈) → 𝐴 ∈ V)
23	22, 22	xpexd 7765	. . . . . 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 17604	. . . . . 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 3577	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) ∧ 𝑣 ∈ 𝑈) → ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ (𝑣 ∩ (𝐴 × 𝐴)))
28		inss1 4133	. . . . . 6 ⊢ (𝑣 ∩ (𝐴 × 𝐴)) ⊆ 𝑣
29		sstr 3916	. . . . . 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 3100	. . . 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 3154	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) → ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
34		iscfilu 24611	. . 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 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3895   ∩ cin 4089  𝒫 cpw 4556   × cxp 5648  ‘cfv 6538  (class class class)co 7420   ↾_t crest 17596  fBascfbas 21671  UnifOncust 24524  CauFil_uccfilu 24609

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-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

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-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-rest 17598  df-fbas 21680  df-ust 24525  df-cfilu 24610

This theorem is used by: (None)