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Theorem metust 24446

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 24445	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝐹 ∈ (fBas‘(𝑋 × 𝑋)))
3		fgcl 23765	. . 3 ⊢ (𝐹 ∈ (fBas‘(𝑋 × 𝑋)) → ((𝑋 × 𝑋)filGen𝐹) ∈ (Fil‘(𝑋 × 𝑋)))
4		filsspw 23738	. . 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 23742	. . 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 730	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) ∧ 𝑣 ⊆ 𝑤) → ((𝑋 × 𝑋)filGen𝐹) ∈ (Fil‘(𝑋 × 𝑋)))
10		simpllr 775	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) ∧ 𝑣 ⊆ 𝑤) → 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹))
11		simplr 768	. . . . . . . 8 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) ∧ 𝑣 ⊆ 𝑤) → 𝑤 ∈ 𝒫 (𝑋 × 𝑋))
12	11	elpwid 4572	. . . . . . 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 23740	. . . . . . 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 1374	. . . . . 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 3125	. . . 4 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)))
18	8	ad2antrr 726	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ((𝑋 × 𝑋)filGen𝐹) ∈ (Fil‘(𝑋 × 𝑋)))
19		simplr 768	. . . . . 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 23741	. . . . . 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 1373	. . . . 5 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)) → (𝑣 ∩ 𝑤) ∈ ((𝑋 × 𝑋)filGen𝐹))
23	22	ralrimiva 3125	. . . 4 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ∀𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)(𝑣 ∩ 𝑤) ∈ ((𝑋 × 𝑋)filGen𝐹))
24	1	metustid 24442	. . . . . . . 8 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑢 ∈ 𝐹) → ( I ↾ 𝑋) ⊆ 𝑢)
25	24	ad5ant24 760	. . . . . . 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 3957	. . . . . 6 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → ( I ↾ 𝑋) ⊆ 𝑣)
28		elfg 23758	. . . . . . . . 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 580	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ∃𝑢 ∈ 𝐹 𝑢 ⊆ 𝑣)
32	27, 31	r19.29a 3141	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ( I ↾ 𝑋) ⊆ 𝑣)
33	8	ad3antrrr 730	. . . . . . 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 23759	. . . . . . . . . 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 726	. . . . . . . 8 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → 𝐹 ⊆ ((𝑋 × 𝑋)filGen𝐹))
38		simplr 768	. . . . . . . 8 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → 𝑢 ∈ 𝐹)
39	37, 38	sseldd 3947	. . . . . . 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 580	. . . . . . . . 9 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → 𝑣 ⊆ (𝑋 × 𝑋))
42	41	ad2antrr 726	. . . . . . . 8 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → 𝑣 ⊆ (𝑋 × 𝑋))
43		cnvss 5836	. . . . . . . . 9 ⊢ (𝑣 ⊆ (𝑋 × 𝑋) → ^◡𝑣 ⊆ ^◡(𝑋 × 𝑋))
44		cnvxp 6130	. . . . . . . . 9 ⊢ ^◡(𝑋 × 𝑋) = (𝑋 × 𝑋)
45	43, 44	sseqtrdi 3987	. . . . . . . 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 24443	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑢 ∈ 𝐹) → ^◡𝑢 = 𝑢)
48	47	ad5ant24 760	. . . . . . . 8 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → ^◡𝑢 = 𝑢)
49		cnvss 5836	. . . . . . . . 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 3982	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → 𝑢 ⊆ ^◡𝑣)
52		filss 23740	. . . . . . 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 1374	. . . . . 6 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → ^◡𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹))
54	53, 31	r19.29a 3141	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ^◡𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹))
55	1	metustexhalf 24444	. . . . . . . . 9 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑢 ∈ 𝐹) → ∃𝑤 ∈ 𝐹 (𝑤 ∘ 𝑤) ⊆ 𝑢)
56	55	ad4ant13 751	. . . . . . . 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 3955	. . . . . . . . . 10 ⊢ (((𝑤 ∘ 𝑤) ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑣) → (𝑤 ∘ 𝑤) ⊆ 𝑣)
59	58	reximi 3067	. . . . . . . . 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 588	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → ∃𝑤 ∈ 𝐹 (𝑤 ∘ 𝑤) ⊆ 𝑣)
62	61, 31	r19.29a 3141	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ∃𝑤 ∈ 𝐹 (𝑤 ∘ 𝑤) ⊆ 𝑣)
63		ssrexv 4016	. . . . . 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 3125	. 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 6896	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
69	68	adantl 481	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝑋 ∈ V)
70		isust 24091	. . 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 1343	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 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 Vcvv 3447 ∩ cin 3913 ⊆ wss 3914 ∅c0 4296 𝒫 cpw 4563 ↦ cmpt 5188 I cid 5532 × cxp 5636 ^◡ccnv 5637 ran crn 5639 ↾ cres 5640 “ cima 5641 ∘ ccom 5642 ‘cfv 6511 (class class class)co 7387 0cc0 11068 ℝ⁺crp 12951 [,)cico 13308 PsMetcpsmet 21248 fBascfbas 21252 filGencfg 21253 Filcfil 23732 UnifOncust 24087

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-er 8671 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ico 13312 df-psmet 21256 df-fbas 21261 df-fg 21262 df-fil 23733 df-ust 24088

This theorem is referenced by: cfilucfil 24447 metuust 24448 metucn 24459

W3C validator