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Theorem metust 23168

Description: The uniform structure generated by a metric 𝐷. (Contributed by Thierry Arnoux, 26-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)

**Hypothesis**
Ref	Expression
metust.1	⊢ 𝐹 = ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑎)))

**Assertion**
Ref	Expression
metust	⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((𝑋 × 𝑋)filGen𝐹) ∈ (UnifOn‘𝑋))

Distinct variable groups: 𝐷,𝑎 𝑋,𝑎 𝐹,𝑎

Proof of Theorem metust Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		metust.1	. . . 4 ⊢ 𝐹 = ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑎)))
2	1	metustfbas 23167	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝐹 ∈ (fBas‘(𝑋 × 𝑋)))
3		fgcl 22486	. . 3 ⊢ (𝐹 ∈ (fBas‘(𝑋 × 𝑋)) → ((𝑋 × 𝑋)filGen𝐹) ∈ (Fil‘(𝑋 × 𝑋)))
4		filsspw 22459	. . 3 ⊢ (((𝑋 × 𝑋)filGen𝐹) ∈ (Fil‘(𝑋 × 𝑋)) → ((𝑋 × 𝑋)filGen𝐹) ⊆ 𝒫 (𝑋 × 𝑋))
5	2, 3, 4	3syl 18	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((𝑋 × 𝑋)filGen𝐹) ⊆ 𝒫 (𝑋 × 𝑋))
6		filtop 22463	. . 3 ⊢ (((𝑋 × 𝑋)filGen𝐹) ∈ (Fil‘(𝑋 × 𝑋)) → (𝑋 × 𝑋) ∈ ((𝑋 × 𝑋)filGen𝐹))
7	2, 3, 6	3syl 18	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑋 × 𝑋) ∈ ((𝑋 × 𝑋)filGen𝐹))
8	2, 3	syl 17	. . . . . . . 8 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((𝑋 × 𝑋)filGen𝐹) ∈ (Fil‘(𝑋 × 𝑋)))
9	8	ad3antrrr 728	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) ∧ 𝑣 ⊆ 𝑤) → ((𝑋 × 𝑋)filGen𝐹) ∈ (Fil‘(𝑋 × 𝑋)))
10		simpllr 774	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) ∧ 𝑣 ⊆ 𝑤) → 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹))
11		simplr 767	. . . . . . . 8 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) ∧ 𝑣 ⊆ 𝑤) → 𝑤 ∈ 𝒫 (𝑋 × 𝑋))
12	11	elpwid 4550	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) ∧ 𝑣 ⊆ 𝑤) → 𝑤 ⊆ (𝑋 × 𝑋))
13		simpr 487	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) ∧ 𝑣 ⊆ 𝑤) → 𝑣 ⊆ 𝑤)
14		filss 22461	. . . . . . 7 ⊢ ((((𝑋 × 𝑋)filGen𝐹) ∈ (Fil‘(𝑋 × 𝑋)) ∧ (𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹) ∧ 𝑤 ⊆ (𝑋 × 𝑋) ∧ 𝑣 ⊆ 𝑤)) → 𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹))
15	9, 10, 12, 13, 14	syl13anc 1368	. . . . . 6 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) ∧ 𝑣 ⊆ 𝑤) → 𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹))
16	15	ex 415	. . . . 5 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) → (𝑣 ⊆ 𝑤 → 𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)))
17	16	ralrimiva 3182	. . . 4 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)))
18	8	ad2antrr 724	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ((𝑋 × 𝑋)filGen𝐹) ∈ (Fil‘(𝑋 × 𝑋)))
19		simplr 767	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)) → 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹))
20		simpr 487	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)) → 𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹))
21		filin 22462	. . . . . 6 ⊢ ((((𝑋 × 𝑋)filGen𝐹) ∈ (Fil‘(𝑋 × 𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹) ∧ 𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)) → (𝑣 ∩ 𝑤) ∈ ((𝑋 × 𝑋)filGen𝐹))
22	18, 19, 20, 21	syl3anc 1367	. . . . 5 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)) → (𝑣 ∩ 𝑤) ∈ ((𝑋 × 𝑋)filGen𝐹))
23	22	ralrimiva 3182	. . . 4 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ∀𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)(𝑣 ∩ 𝑤) ∈ ((𝑋 × 𝑋)filGen𝐹))
24	1	metustid 23164	. . . . . . . 8 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑢 ∈ 𝐹) → ( I ↾ 𝑋) ⊆ 𝑢)
25	24	ad5ant24 759	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → ( I ↾ 𝑋) ⊆ 𝑢)
26		simpr 487	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → 𝑢 ⊆ 𝑣)
27	25, 26	sstrd 3977	. . . . . 6 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → ( I ↾ 𝑋) ⊆ 𝑣)
28		elfg 22479	. . . . . . . . 9 ⊢ (𝐹 ∈ (fBas‘(𝑋 × 𝑋)) → (𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹) ↔ (𝑣 ⊆ (𝑋 × 𝑋) ∧ ∃𝑢 ∈ 𝐹 𝑢 ⊆ 𝑣)))
29	28	biimpa 479	. . . . . . . 8 ⊢ ((𝐹 ∈ (fBas‘(𝑋 × 𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → (𝑣 ⊆ (𝑋 × 𝑋) ∧ ∃𝑢 ∈ 𝐹 𝑢 ⊆ 𝑣))
30	29	simprd 498	. . . . . . 7 ⊢ ((𝐹 ∈ (fBas‘(𝑋 × 𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ∃𝑢 ∈ 𝐹 𝑢 ⊆ 𝑣)
31	2, 30	sylan 582	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ∃𝑢 ∈ 𝐹 𝑢 ⊆ 𝑣)
32	27, 31	r19.29a 3289	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ( I ↾ 𝑋) ⊆ 𝑣)
33	8	ad3antrrr 728	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → ((𝑋 × 𝑋)filGen𝐹) ∈ (Fil‘(𝑋 × 𝑋)))
34	2	adantr 483	. . . . . . . . . 10 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → 𝐹 ∈ (fBas‘(𝑋 × 𝑋)))
35		ssfg 22480	. . . . . . . . . 10 ⊢ (𝐹 ∈ (fBas‘(𝑋 × 𝑋)) → 𝐹 ⊆ ((𝑋 × 𝑋)filGen𝐹))
36	34, 35	syl 17	. . . . . . . . 9 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → 𝐹 ⊆ ((𝑋 × 𝑋)filGen𝐹))
37	36	ad2antrr 724	. . . . . . . 8 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → 𝐹 ⊆ ((𝑋 × 𝑋)filGen𝐹))
38		simplr 767	. . . . . . . 8 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → 𝑢 ∈ 𝐹)
39	37, 38	sseldd 3968	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → 𝑢 ∈ ((𝑋 × 𝑋)filGen𝐹))
40	29	simpld 497	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (fBas‘(𝑋 × 𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → 𝑣 ⊆ (𝑋 × 𝑋))
41	2, 40	sylan 582	. . . . . . . . 9 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → 𝑣 ⊆ (𝑋 × 𝑋))
42	41	ad2antrr 724	. . . . . . . 8 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → 𝑣 ⊆ (𝑋 × 𝑋))
43		cnvss 5743	. . . . . . . . 9 ⊢ (𝑣 ⊆ (𝑋 × 𝑋) → ^◡𝑣 ⊆ ^◡(𝑋 × 𝑋))
44		cnvxp 6014	. . . . . . . . 9 ⊢ ^◡(𝑋 × 𝑋) = (𝑋 × 𝑋)
45	43, 44	sseqtrdi 4017	. . . . . . . 8 ⊢ (𝑣 ⊆ (𝑋 × 𝑋) → ^◡𝑣 ⊆ (𝑋 × 𝑋))
46	42, 45	syl 17	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → ^◡𝑣 ⊆ (𝑋 × 𝑋))
47	1	metustsym 23165	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑢 ∈ 𝐹) → ^◡𝑢 = 𝑢)
48	47	ad5ant24 759	. . . . . . . 8 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → ^◡𝑢 = 𝑢)
49		cnvss 5743	. . . . . . . . 9 ⊢ (𝑢 ⊆ 𝑣 → ^◡𝑢 ⊆ ^◡𝑣)
50	49	adantl 484	. . . . . . . 8 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → ^◡𝑢 ⊆ ^◡𝑣)
51	48, 50	eqsstrrd 4006	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → 𝑢 ⊆ ^◡𝑣)
52		filss 22461	. . . . . . 7 ⊢ ((((𝑋 × 𝑋)filGen𝐹) ∈ (Fil‘(𝑋 × 𝑋)) ∧ (𝑢 ∈ ((𝑋 × 𝑋)filGen𝐹) ∧ ^◡𝑣 ⊆ (𝑋 × 𝑋) ∧ 𝑢 ⊆ ^◡𝑣)) → ^◡𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹))
53	33, 39, 46, 51, 52	syl13anc 1368	. . . . . 6 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → ^◡𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹))
54	53, 31	r19.29a 3289	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ^◡𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹))
55	1	metustexhalf 23166	. . . . . . . . 9 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑢 ∈ 𝐹) → ∃𝑤 ∈ 𝐹 (𝑤 ∘ 𝑤) ⊆ 𝑢)
56	55	ad4ant13 749	. . . . . . . 8 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → ∃𝑤 ∈ 𝐹 (𝑤 ∘ 𝑤) ⊆ 𝑢)
57		r19.41v 3347	. . . . . . . . 9 ⊢ (∃𝑤 ∈ 𝐹 ((𝑤 ∘ 𝑤) ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑣) ↔ (∃𝑤 ∈ 𝐹 (𝑤 ∘ 𝑤) ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑣))
58		sstr 3975	. . . . . . . . . 10 ⊢ (((𝑤 ∘ 𝑤) ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑣) → (𝑤 ∘ 𝑤) ⊆ 𝑣)
59	58	reximi 3243	. . . . . . . . 9 ⊢ (∃𝑤 ∈ 𝐹 ((𝑤 ∘ 𝑤) ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑣) → ∃𝑤 ∈ 𝐹 (𝑤 ∘ 𝑤) ⊆ 𝑣)
60	57, 59	sylbir 237	. . . . . . . 8 ⊢ ((∃𝑤 ∈ 𝐹 (𝑤 ∘ 𝑤) ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑣) → ∃𝑤 ∈ 𝐹 (𝑤 ∘ 𝑤) ⊆ 𝑣)
61	56, 60	sylancom 590	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → ∃𝑤 ∈ 𝐹 (𝑤 ∘ 𝑤) ⊆ 𝑣)
62	61, 31	r19.29a 3289	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ∃𝑤 ∈ 𝐹 (𝑤 ∘ 𝑤) ⊆ 𝑣)
63		ssrexv 4034	. . . . . 6 ⊢ (𝐹 ⊆ ((𝑋 × 𝑋)filGen𝐹) → (∃𝑤 ∈ 𝐹 (𝑤 ∘ 𝑤) ⊆ 𝑣 → ∃𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)(𝑤 ∘ 𝑤) ⊆ 𝑣))
64	36, 62, 63	sylc 65	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ∃𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)(𝑤 ∘ 𝑤) ⊆ 𝑣)
65	32, 54, 64	3jca 1124	. . . 4 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → (( I ↾ 𝑋) ⊆ 𝑣 ∧ ^◡𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹) ∧ ∃𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)(𝑤 ∘ 𝑤) ⊆ 𝑣))
66	17, 23, 65	3jca 1124	. . 3 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ ∀𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)(𝑣 ∩ 𝑤) ∈ ((𝑋 × 𝑋)filGen𝐹) ∧ (( I ↾ 𝑋) ⊆ 𝑣 ∧ ^◡𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹) ∧ ∃𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)(𝑤 ∘ 𝑤) ⊆ 𝑣)))
67	66	ralrimiva 3182	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ∀𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)(∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ ∀𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)(𝑣 ∩ 𝑤) ∈ ((𝑋 × 𝑋)filGen𝐹) ∧ (( I ↾ 𝑋) ⊆ 𝑣 ∧ ^◡𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹) ∧ ∃𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)(𝑤 ∘ 𝑤) ⊆ 𝑣)))
68		elfvex 6703	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
69	68	adantl 484	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝑋 ∈ V)
70		isust 22812	. . 3 ⊢ (𝑋 ∈ V → (((𝑋 × 𝑋)filGen𝐹) ∈ (UnifOn‘𝑋) ↔ (((𝑋 × 𝑋)filGen𝐹) ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ ((𝑋 × 𝑋)filGen𝐹) ∧ ∀𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)(∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ ∀𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)(𝑣 ∩ 𝑤) ∈ ((𝑋 × 𝑋)filGen𝐹) ∧ (( I ↾ 𝑋) ⊆ 𝑣 ∧ ^◡𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹) ∧ ∃𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)(𝑤 ∘ 𝑤) ⊆ 𝑣)))))
71	69, 70	syl 17	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (((𝑋 × 𝑋)filGen𝐹) ∈ (UnifOn‘𝑋) ↔ (((𝑋 × 𝑋)filGen𝐹) ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ ((𝑋 × 𝑋)filGen𝐹) ∧ ∀𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)(∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ ∀𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)(𝑣 ∩ 𝑤) ∈ ((𝑋 × 𝑋)filGen𝐹) ∧ (( I ↾ 𝑋) ⊆ 𝑣 ∧ ^◡𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹) ∧ ∃𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)(𝑤 ∘ 𝑤) ⊆ 𝑣)))))
72	5, 7, 67, 71	mpbir3and 1338	1 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((𝑋 × 𝑋)filGen𝐹) ∈ (UnifOn‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 𝒫 cpw 4539 ↦ cmpt 5146 I cid 5459 × cxp 5553 ^◡ccnv 5554 ran crn 5556 ↾ cres 5557 “ cima 5558 ∘ ccom 5559 ‘cfv 6355 (class class class)co 7156 0cc0 10537 ℝ⁺crp 12390 [,)cico 12741 PsMetcpsmet 20529 fBascfbas 20533 filGencfg 20534 Filcfil 22453 UnifOncust 22808

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-2 11701 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ico 12745 df-psmet 20537 df-fbas 20542 df-fg 20543 df-fil 22454 df-ust 22809

This theorem is referenced by: cfilucfil 23169 metuust 23170 metucn 23181

W3C validator