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Theorem metust 24466

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 24465	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝐹 ∈ (fBas‘(𝑋 × 𝑋)))
3		fgcl 23786	. . 3 ⊢ (𝐹 ∈ (fBas‘(𝑋 × 𝑋)) → ((𝑋 × 𝑋)filGen𝐹) ∈ (Fil‘(𝑋 × 𝑋)))
4		filsspw 23759	. . 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 23763	. . 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 4557	. . . . . . 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 23761	. . . . . . 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 3122	. . . 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 23762	. . . . . 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 3122	. . . 4 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ∀𝑤 ∈ ((𝑋 × 𝑋)filGen𝐹)(𝑣 ∩ 𝑤) ∈ ((𝑋 × 𝑋)filGen𝐹))
24	1	metustid 24462	. . . . . . . 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 3943	. . . . . 6 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → ( I ↾ 𝑋) ⊆ 𝑣)
28		elfg 23779	. . . . . . . . 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 3138	. . . . 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 23780	. . . . . . . . . 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 3933	. . . . . . 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 5810	. . . . . . . . 9 ⊢ (𝑣 ⊆ (𝑋 × 𝑋) → ^◡𝑣 ⊆ ^◡(𝑋 × 𝑋))
44		cnvxp 6101	. . . . . . . . 9 ⊢ ^◡(𝑋 × 𝑋) = (𝑋 × 𝑋)
45	43, 44	sseqtrdi 3973	. . . . . . . 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 24463	. . . . . . . . 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 5810	. . . . . . . . 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 3968	. . . . . . 7 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑢 ∈ 𝐹) ∧ 𝑢 ⊆ 𝑣) → 𝑢 ⊆ ^◡𝑣)
52		filss 23761	. . . . . . 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 3138	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ^◡𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹))
55	1	metustexhalf 24464	. . . . . . . . 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 3160	. . . . . . . . 9 ⊢ (∃𝑤 ∈ 𝐹 ((𝑤 ∘ 𝑤) ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑣) ↔ (∃𝑤 ∈ 𝐹 (𝑤 ∘ 𝑤) ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑣))
58		sstr 3941	. . . . . . . . . 10 ⊢ (((𝑤 ∘ 𝑤) ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑣) → (𝑤 ∘ 𝑤) ⊆ 𝑣)
59	58	reximi 3068	. . . . . . . . 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 3138	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ∃𝑤 ∈ 𝐹 (𝑤 ∘ 𝑤) ⊆ 𝑣)
63		ssrexv 4002	. . . . . 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 3122	. 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 6852	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
69	68	adantl 481	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝑋 ∈ V)
70		isust 24112	. . 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 1541 ∈ wcel 2110 ≠ wne 2926 ∀wral 3045 ∃wrex 3054 Vcvv 3434 ∩ cin 3899 ⊆ wss 3900 ∅c0 4281 𝒫 cpw 4548 ↦ cmpt 5170 I cid 5508 × cxp 5612 ^◡ccnv 5613 ran crn 5615 ↾ cres 5616 “ cima 5617 ∘ ccom 5618 ‘cfv 6477 (class class class)co 7341 0cc0 10998 ℝ⁺crp 12882 [,)cico 13239 PsMetcpsmet 21268 fBascfbas 21272 filGencfg 21273 Filcfil 23753 UnifOncust 24108

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 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-cnex 11054 ax-resscn 11055 ax-1cn 11056 ax-icn 11057 ax-addcl 11058 ax-addrcl 11059 ax-mulcl 11060 ax-mulrcl 11061 ax-mulcom 11062 ax-addass 11063 ax-mulass 11064 ax-distr 11065 ax-i2m1 11066 ax-1ne0 11067 ax-1rid 11068 ax-rnegex 11069 ax-rrecex 11070 ax-cnre 11071 ax-pre-lttri 11072 ax-pre-lttrn 11073 ax-pre-ltadd 11074 ax-pre-mulgt0 11075

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3344 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4858 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6244 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-er 8617 df-map 8747 df-en 8865 df-dom 8866 df-sdom 8867 df-pnf 11140 df-mnf 11141 df-xr 11142 df-ltxr 11143 df-le 11144 df-sub 11338 df-neg 11339 df-div 11767 df-nn 12118 df-2 12180 df-rp 12883 df-xneg 13003 df-xadd 13004 df-xmul 13005 df-ico 13243 df-psmet 21276 df-fbas 21281 df-fg 21282 df-fil 23754 df-ust 24109

This theorem is referenced by: cfilucfil 24467 metuust 24468 metucn 24479

W3C validator