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Theorem metust 24066

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 24065	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝐹 ∈ (fBas‘(𝑋 × 𝑋)))
3		fgcl 23381	. . 3 ⊢ (𝐹 ∈ (fBas‘(𝑋 × 𝑋)) → ((𝑋 × 𝑋)filGen𝐹) ∈ (Fil‘(𝑋 × 𝑋)))
4		filsspw 23354	. . 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 23358	. . 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 4611	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) ∧ 𝑣 ⊆ 𝑤) → 𝑤 ⊆ (𝑋 × 𝑋))
13		simpr 485	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) ∧ 𝑣 ⊆ 𝑤) → 𝑣 ⊆ 𝑤)
14		filss 23356	. . . . . . 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 1372	. . . . . 6 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) ∧ 𝑣 ⊆ 𝑤) → 𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹))
16	15	ex 413	. . . . 5 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) → (𝑣 ⊆ 𝑤 → 𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)))
17	16	ralrimiva 3146	. . . 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 485	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)) → 𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹))
21		filin 23357	. . . . . 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 1371	. . . . 5 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)) → (𝑣 ∩ 𝑤) ∈ ((𝑋 × 𝑋)filGen𝐹))
23	22	ralrimiva 3146	. . . 4 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ∀𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)(𝑣 ∩ 𝑤) ∈ ((𝑋 × 𝑋)filGen𝐹))
24	1	metustid 24062	. . . . . . . 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 485	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → 𝑢 ⊆ 𝑣)
27	25, 26	sstrd 3992	. . . . . 6 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → ( I ↾ 𝑋) ⊆ 𝑣)
28		elfg 23374	. . . . . . . . 9 ⊢ (𝐹 ∈ (fBas‘(𝑋 × 𝑋)) → (𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹) ↔ (𝑣 ⊆ (𝑋 × 𝑋) ∧ ∃𝑢 ∈ 𝐹 𝑢 ⊆ 𝑣)))
29	28	biimpa 477	. . . . . . . 8 ⊢ ((𝐹 ∈ (fBas‘(𝑋 × 𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → (𝑣 ⊆ (𝑋 × 𝑋) ∧ ∃𝑢 ∈ 𝐹 𝑢 ⊆ 𝑣))
30	29	simprd 496	. . . . . . 7 ⊢ ((𝐹 ∈ (fBas‘(𝑋 × 𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ∃𝑢 ∈ 𝐹 𝑢 ⊆ 𝑣)
31	2, 30	sylan 580	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ∃𝑢 ∈ 𝐹 𝑢 ⊆ 𝑣)
32	27, 31	r19.29a 3162	. . . . 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 481	. . . . . . . . . 10 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → 𝐹 ∈ (fBas‘(𝑋 × 𝑋)))
35		ssfg 23375	. . . . . . . . . 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 3983	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → 𝑢 ∈ ((𝑋 × 𝑋)filGen𝐹))
40	29	simpld 495	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (fBas‘(𝑋 × 𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → 𝑣 ⊆ (𝑋 × 𝑋))
41	2, 40	sylan 580	. . . . . . . . 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 5872	. . . . . . . . 9 ⊢ (𝑣 ⊆ (𝑋 × 𝑋) → ^◡𝑣 ⊆ ^◡(𝑋 × 𝑋))
44		cnvxp 6156	. . . . . . . . 9 ⊢ ^◡(𝑋 × 𝑋) = (𝑋 × 𝑋)
45	43, 44	sseqtrdi 4032	. . . . . . . 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 24063	. . . . . . . . 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 5872	. . . . . . . . 9 ⊢ (𝑢 ⊆ 𝑣 → ^◡𝑢 ⊆ ^◡𝑣)
50	49	adantl 482	. . . . . . . 8 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → ^◡𝑢 ⊆ ^◡𝑣)
51	48, 50	eqsstrrd 4021	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → 𝑢 ⊆ ^◡𝑣)
52		filss 23356	. . . . . . 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 1372	. . . . . 6 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → ^◡𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹))
54	53, 31	r19.29a 3162	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ^◡𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹))
55	1	metustexhalf 24064	. . . . . . . . 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 3188	. . . . . . . . 9 ⊢ (∃𝑤 ∈ 𝐹 ((𝑤 ∘ 𝑤) ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑣) ↔ (∃𝑤 ∈ 𝐹 (𝑤 ∘ 𝑤) ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑣))
58		sstr 3990	. . . . . . . . . 10 ⊢ (((𝑤 ∘ 𝑤) ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑣) → (𝑤 ∘ 𝑤) ⊆ 𝑣)
59	58	reximi 3084	. . . . . . . . 9 ⊢ (∃𝑤 ∈ 𝐹 ((𝑤 ∘ 𝑤) ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑣) → ∃𝑤 ∈ 𝐹 (𝑤 ∘ 𝑤) ⊆ 𝑣)
60	57, 59	sylbir 234	. . . . . . . 8 ⊢ ((∃𝑤 ∈ 𝐹 (𝑤 ∘ 𝑤) ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑣) → ∃𝑤 ∈ 𝐹 (𝑤 ∘ 𝑤) ⊆ 𝑣)
61	56, 60	sylancom 588	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → ∃𝑤 ∈ 𝐹 (𝑤 ∘ 𝑤) ⊆ 𝑣)
62	61, 31	r19.29a 3162	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ∃𝑤 ∈ 𝐹 (𝑤 ∘ 𝑤) ⊆ 𝑣)
63		ssrexv 4051	. . . . . 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 1128	. . . 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 1128	. . 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 3146	. 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 6929	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
69	68	adantl 482	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝑋 ∈ V)
70		isust 23707	. . 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 1342	1 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((𝑋 × 𝑋)filGen𝐹) ∈ (UnifOn‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 Vcvv 3474 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 𝒫 cpw 4602 ↦ cmpt 5231 I cid 5573 × cxp 5674 ^◡ccnv 5675 ran crn 5677 ↾ cres 5678 “ cima 5679 ∘ ccom 5680 ‘cfv 6543 (class class class)co 7408 0cc0 11109 ℝ⁺crp 12973 [,)cico 13325 PsMetcpsmet 20927 fBascfbas 20931 filGencfg 20932 Filcfil 23348 UnifOncust 23703

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-2 12274 df-rp 12974 df-xneg 13091 df-xadd 13092 df-xmul 13093 df-ico 13329 df-psmet 20935 df-fbas 20940 df-fg 20941 df-fil 23349 df-ust 23704

This theorem is referenced by: cfilucfil 24067 metuust 24068 metucn 24079

W3C validator