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Theorem metust 24506

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 24505	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝐹 ∈ (fBas‘(𝑋 × 𝑋)))
3		fgcl 23826	. . 3 ⊢ (𝐹 ∈ (fBas‘(𝑋 × 𝑋)) → ((𝑋 × 𝑋)filGen𝐹) ∈ (Fil‘(𝑋 × 𝑋)))
4		filsspw 23799	. . 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 23803	. . 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 731	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) ∧ 𝑣 ⊆ 𝑤) → ((𝑋 × 𝑋)filGen𝐹) ∈ (Fil‘(𝑋 × 𝑋)))
10		simpllr 776	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) ∧ 𝑣 ⊆ 𝑤) → 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹))
11		simplr 769	. . . . . . . 8 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) ∧ 𝑣 ⊆ 𝑤) → 𝑤 ∈ 𝒫 (𝑋 × 𝑋))
12	11	elpwid 4564	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) ∧ 𝑣 ⊆ 𝑤) → 𝑤 ⊆ (𝑋 × 𝑋))
13		simpr 484	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) ∧ 𝑣 ⊆ 𝑤) → 𝑣 ⊆ 𝑤)
14		filss 23801	. . . . . . 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 1375	. . . . . 6 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) ∧ 𝑣 ⊆ 𝑤) → 𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹))
16	15	ex 412	. . . . 5 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) → (𝑣 ⊆ 𝑤 → 𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)))
17	16	ralrimiva 3129	. . . 4 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)))
18	8	ad2antrr 727	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ((𝑋 × 𝑋)filGen𝐹) ∈ (Fil‘(𝑋 × 𝑋)))
19		simplr 769	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)) → 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹))
20		simpr 484	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)) → 𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹))
21		filin 23802	. . . . . 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 1374	. . . . 5 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)) → (𝑣 ∩ 𝑤) ∈ ((𝑋 × 𝑋)filGen𝐹))
23	22	ralrimiva 3129	. . . 4 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ∀𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)(𝑣 ∩ 𝑤) ∈ ((𝑋 × 𝑋)filGen𝐹))
24	1	metustid 24502	. . . . . . . 8 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑢 ∈ 𝐹) → ( I ↾ 𝑋) ⊆ 𝑢)
25	24	ad5ant24 761	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → ( I ↾ 𝑋) ⊆ 𝑢)
26		simpr 484	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → 𝑢 ⊆ 𝑣)
27	25, 26	sstrd 3945	. . . . . 6 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → ( I ↾ 𝑋) ⊆ 𝑣)
28		elfg 23819	. . . . . . . . 9 ⊢ (𝐹 ∈ (fBas‘(𝑋 × 𝑋)) → (𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹) ↔ (𝑣 ⊆ (𝑋 × 𝑋) ∧ ∃𝑢 ∈ 𝐹 𝑢 ⊆ 𝑣)))
29	28	biimpa 476	. . . . . . . 8 ⊢ ((𝐹 ∈ (fBas‘(𝑋 × 𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → (𝑣 ⊆ (𝑋 × 𝑋) ∧ ∃𝑢 ∈ 𝐹 𝑢 ⊆ 𝑣))
30	29	simprd 495	. . . . . . 7 ⊢ ((𝐹 ∈ (fBas‘(𝑋 × 𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ∃𝑢 ∈ 𝐹 𝑢 ⊆ 𝑣)
31	2, 30	sylan 581	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ∃𝑢 ∈ 𝐹 𝑢 ⊆ 𝑣)
32	27, 31	r19.29a 3145	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ( I ↾ 𝑋) ⊆ 𝑣)
33	8	ad3antrrr 731	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → ((𝑋 × 𝑋)filGen𝐹) ∈ (Fil‘(𝑋 × 𝑋)))
34	2	adantr 480	. . . . . . . . . 10 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → 𝐹 ∈ (fBas‘(𝑋 × 𝑋)))
35		ssfg 23820	. . . . . . . . . 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 727	. . . . . . . 8 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → 𝐹 ⊆ ((𝑋 × 𝑋)filGen𝐹))
38		simplr 769	. . . . . . . 8 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → 𝑢 ∈ 𝐹)
39	37, 38	sseldd 3935	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → 𝑢 ∈ ((𝑋 × 𝑋)filGen𝐹))
40	29	simpld 494	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (fBas‘(𝑋 × 𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → 𝑣 ⊆ (𝑋 × 𝑋))
41	2, 40	sylan 581	. . . . . . . . 9 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → 𝑣 ⊆ (𝑋 × 𝑋))
42	41	ad2antrr 727	. . . . . . . 8 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → 𝑣 ⊆ (𝑋 × 𝑋))
43		cnvss 5822	. . . . . . . . 9 ⊢ (𝑣 ⊆ (𝑋 × 𝑋) → ^◡𝑣 ⊆ ^◡(𝑋 × 𝑋))
44		cnvxp 6116	. . . . . . . . 9 ⊢ ^◡(𝑋 × 𝑋) = (𝑋 × 𝑋)
45	43, 44	sseqtrdi 3975	. . . . . . . 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 24503	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑢 ∈ 𝐹) → ^◡𝑢 = 𝑢)
48	47	ad5ant24 761	. . . . . . . 8 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → ^◡𝑢 = 𝑢)
49		cnvss 5822	. . . . . . . . 9 ⊢ (𝑢 ⊆ 𝑣 → ^◡𝑢 ⊆ ^◡𝑣)
50	49	adantl 481	. . . . . . . 8 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → ^◡𝑢 ⊆ ^◡𝑣)
51	48, 50	eqsstrrd 3970	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → 𝑢 ⊆ ^◡𝑣)
52		filss 23801	. . . . . . 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 1375	. . . . . 6 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → ^◡𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹))
54	53, 31	r19.29a 3145	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ^◡𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹))
55	1	metustexhalf 24504	. . . . . . . . 9 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑢 ∈ 𝐹) → ∃𝑤 ∈ 𝐹 (𝑤 ∘ 𝑤) ⊆ 𝑢)
56	55	ad4ant13 752	. . . . . . . 8 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → ∃𝑤 ∈ 𝐹 (𝑤 ∘ 𝑤) ⊆ 𝑢)
57		r19.41v 3167	. . . . . . . . 9 ⊢ (∃𝑤 ∈ 𝐹 ((𝑤 ∘ 𝑤) ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑣) ↔ (∃𝑤 ∈ 𝐹 (𝑤 ∘ 𝑤) ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑣))
58		sstr 3943	. . . . . . . . . 10 ⊢ (((𝑤 ∘ 𝑤) ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑣) → (𝑤 ∘ 𝑤) ⊆ 𝑣)
59	58	reximi 3075	. . . . . . . . 9 ⊢ (∃𝑤 ∈ 𝐹 ((𝑤 ∘ 𝑤) ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑣) → ∃𝑤 ∈ 𝐹 (𝑤 ∘ 𝑤) ⊆ 𝑣)
60	57, 59	sylbir 235	. . . . . . . 8 ⊢ ((∃𝑤 ∈ 𝐹 (𝑤 ∘ 𝑤) ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑣) → ∃𝑤 ∈ 𝐹 (𝑤 ∘ 𝑤) ⊆ 𝑣)
61	56, 60	sylancom 589	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → ∃𝑤 ∈ 𝐹 (𝑤 ∘ 𝑤) ⊆ 𝑣)
62	61, 31	r19.29a 3145	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ∃𝑤 ∈ 𝐹 (𝑤 ∘ 𝑤) ⊆ 𝑣)
63		ssrexv 4004	. . . . . 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 1129	. . . 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 1129	. . 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 3129	. 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 6870	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
69	68	adantl 481	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝑋 ∈ V)
70		isust 24152	. . 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 1344	1 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((𝑋 × 𝑋)filGen𝐹) ∈ (UnifOn‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∃wrex 3061 Vcvv 3441 ∩ cin 3901 ⊆ wss 3902 ∅c0 4286 𝒫 cpw 4555 ↦ cmpt 5180 I cid 5519 × cxp 5623 ^◡ccnv 5624 ran crn 5626 ↾ cres 5627 “ cima 5628 ∘ ccom 5629 ‘cfv 6493 (class class class)co 7360 0cc0 11030 ℝ⁺crp 12909 [,)cico 13267 PsMetcpsmet 21297 fBascfbas 21301 filGencfg 21302 Filcfil 23793 UnifOncust 24148

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-er 8637 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12150 df-2 12212 df-rp 12910 df-xneg 13030 df-xadd 13031 df-xmul 13032 df-ico 13271 df-psmet 21305 df-fbas 21310 df-fg 21311 df-fil 23794 df-ust 24149

This theorem is referenced by: cfilucfil 24507 metuust 24508 metucn 24519

W3C validator