Theorem psmetutop 24692

Description: The topology induced by a uniform structure generated by a metric 𝐷 is generated by that metric's open balls. (Contributed by Thierry Arnoux, 6-Dec-2017.) (Revised by Thierry Arnoux, 11-Mar-2018.)

Assertion

Ref	Expression
psmetutop	⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (unifTop‘(metUnif‘𝐷)) = (topGen‘ran (ball‘𝐷)))

Proof of Theorem psmetutop

Dummy variables 𝑎 𝑏 𝑑 𝑒 𝑣 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		metuust 24685	. . . . . . . . . . . 12 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (metUnif‘𝐷) ∈ (UnifOn‘𝑋))
2		utopval 24357	. . . . . . . . . . . 12 ⊢ ((metUnif‘𝐷) ∈ (UnifOn‘𝑋) → (unifTop‘(metUnif‘𝐷)) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎})
3	1, 2	syl 18	. . . . . . . . . . 11 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (unifTop‘(metUnif‘𝐷)) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎})
4	3	eleq2d 2855	. . . . . . . . . 10 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑎 ∈ (unifTop‘(metUnif‘𝐷)) ↔ 𝑎 ∈ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎}))
5		rabid 3444	. . . . . . . . . 10 ⊢ (𝑎 ∈ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎} ↔ (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎))
6	4, 5	bitrdi 290	. . . . . . . . 9 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑎 ∈ (unifTop‘(metUnif‘𝐷)) ↔ (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)))
7	6	biimpa 481	. . . . . . . 8 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎))
8	7	simpld 499	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → 𝑎 ∈ 𝒫 𝑋)
9	8	elpwid 4576	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → 𝑎 ⊆ 𝑋)
10		unirnblps 24544	. . . . . . 7 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∪ ran (ball‘𝐷) = 𝑋)
11	10	ad2antlr 739	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → ∪ ran (ball‘𝐷) = 𝑋)
12	9, 11	sseqtrrd 3982	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → 𝑎 ⊆ ∪ ran (ball‘𝐷))
13		simpr 489	. . . . . . . 8 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → (𝑣 “ {𝑥}) ⊆ 𝑎)
14		simp-5r 797	. . . . . . . . 9 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝐷 ∈ (PsMet‘𝑋))
15		simplr 780	. . . . . . . . 9 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑣 ∈ (metUnif‘𝐷))
16	9	ad3antrrr 742	. . . . . . . . . 10 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑎 ⊆ 𝑋)
17		simpllr 787	. . . . . . . . . 10 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑥 ∈ 𝑎)
18	16, 17	sseldd 3946	. . . . . . . . 9 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑥 ∈ 𝑋)
19		metustbl 24691	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷) ∧ 𝑥 ∈ 𝑋) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ (𝑣 “ {𝑥})))
20	14, 15, 18, 19	syl3anc 1396	. . . . . . . 8 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ (𝑣 “ {𝑥})))
21		sstr 3953	. . . . . . . . . . 11 ⊢ ((𝑏 ⊆ (𝑣 “ {𝑥}) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑏 ⊆ 𝑎)
22	21	expcom 418	. . . . . . . . . 10 ⊢ ((𝑣 “ {𝑥}) ⊆ 𝑎 → (𝑏 ⊆ (𝑣 “ {𝑥}) → 𝑏 ⊆ 𝑎))
23	22	anim2d 623	. . . . . . . . 9 ⊢ ((𝑣 “ {𝑥}) ⊆ 𝑎 → ((𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ (𝑣 “ {𝑥})) → (𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑎)))
24	23	reximdv 3186	. . . . . . . 8 ⊢ ((𝑣 “ {𝑥}) ⊆ 𝑎 → (∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ (𝑣 “ {𝑥})) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑎)))
25	13, 20, 24	sylc 66	. . . . . . 7 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑎))
26	7	simprd 500	. . . . . . . 8 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)
27	26	r19.21bi 3263	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) → ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)
28	25, 27	r19.29a 3179	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑎))
29	28	ralrimiva 3163	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → ∀𝑥 ∈ 𝑎 ∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑎))
30	12, 29	jca 520	. . . 4 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → (𝑎 ⊆ ∪ ran (ball‘𝐷) ∧ ∀𝑥 ∈ 𝑎 ∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑎)))
31		fvex 6895	. . . . . 6 ⊢ (ball‘𝐷) ∈ V
32	31	rnex 7906	. . . . 5 ⊢ ran (ball‘𝐷) ∈ V
33		eltg2 23083	. . . . 5 ⊢ (ran (ball‘𝐷) ∈ V → (𝑎 ∈ (topGen‘ran (ball‘𝐷)) ↔ (𝑎 ⊆ ∪ ran (ball‘𝐷) ∧ ∀𝑥 ∈ 𝑎 ∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑎))))
34	32, 33	mp1i 14	. . . 4 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → (𝑎 ∈ (topGen‘ran (ball‘𝐷)) ↔ (𝑎 ⊆ ∪ ran (ball‘𝐷) ∧ ∀𝑥 ∈ 𝑎 ∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑎))))
35	30, 34	mpbird 260	. . 3 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → 𝑎 ∈ (topGen‘ran (ball‘𝐷)))
36	32, 33	mp1i 14	. . . . . . . . 9 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑎 ∈ (topGen‘ran (ball‘𝐷)) ↔ (𝑎 ⊆ ∪ ran (ball‘𝐷) ∧ ∀𝑥 ∈ 𝑎 ∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑎))))
37	36	biimpa 481	. . . . . . . 8 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → (𝑎 ⊆ ∪ ran (ball‘𝐷) ∧ ∀𝑥 ∈ 𝑎 ∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑎)))
38	37	simpld 499	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎 ⊆ ∪ ran (ball‘𝐷))
39	10	ad2antlr 739	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → ∪ ran (ball‘𝐷) = 𝑋)
40	38, 39	sseqtrd 3981	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎 ⊆ 𝑋)
41		elpwg 4570	. . . . . . 7 ⊢ (𝑎 ∈ (topGen‘ran (ball‘𝐷)) → (𝑎 ∈ 𝒫 𝑋 ↔ 𝑎 ⊆ 𝑋))
42	41	adantl 486	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → (𝑎 ∈ 𝒫 𝑋 ↔ 𝑎 ⊆ 𝑋))
43	40, 42	mpbird 260	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎 ∈ 𝒫 𝑋)
44		simpllr 787	. . . . . . . . 9 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥 ∈ 𝑎) → 𝐷 ∈ (PsMet‘𝑋))
45	40	sselda 3945	. . . . . . . . 9 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥 ∈ 𝑎) → 𝑥 ∈ 𝑋)
46	37	simprd 500	. . . . . . . . . . 11 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → ∀𝑥 ∈ 𝑎 ∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑎))
47	46	r19.21bi 3263	. . . . . . . . . 10 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥 ∈ 𝑎) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑎))
48		blssexps 24551	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋) → (∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑎) ↔ ∃𝑑 ∈ ℝ+ (𝑥(ball‘𝐷)𝑑) ⊆ 𝑎))
49	44, 45, 48	syl2anc 595	. . . . . . . . . 10 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥 ∈ 𝑎) → (∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑎) ↔ ∃𝑑 ∈ ℝ+ (𝑥(ball‘𝐷)𝑑) ⊆ 𝑎))
50	47, 49	mpbid 235	. . . . . . . . 9 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥 ∈ 𝑎) → ∃𝑑 ∈ ℝ+ (𝑥(ball‘𝐷)𝑑) ⊆ 𝑎)
51		blval2 24687	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑑 ∈ ℝ+) → (𝑥(ball‘𝐷)𝑑) = ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}))
52	51	3expa 1134	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑑 ∈ ℝ+) → (𝑥(ball‘𝐷)𝑑) = ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}))
53	52	sseq1d 3976	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑑 ∈ ℝ+) → ((𝑥(ball‘𝐷)𝑑) ⊆ 𝑎 ↔ ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
54	53	rexbidva 3193	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋) → (∃𝑑 ∈ ℝ+ (𝑥(ball‘𝐷)𝑑) ⊆ 𝑎 ↔ ∃𝑑 ∈ ℝ+ ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
55	54	biimpa 481	. . . . . . . . 9 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ ∃𝑑 ∈ ℝ+ (𝑥(ball‘𝐷)𝑑) ⊆ 𝑎) → ∃𝑑 ∈ ℝ+ ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎)
56	44, 45, 50, 55	syl21anc 850	. . . . . . . 8 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥 ∈ 𝑎) → ∃𝑑 ∈ ℝ+ ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎)
57		cnvexg 7920	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ◡𝐷 ∈ V)
58		imaexg 7909	. . . . . . . . . . 11 ⊢ (◡𝐷 ∈ V → (◡𝐷 “ (0[,)𝑑)) ∈ V)
59	57, 58	syl 18	. . . . . . . . . 10 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (◡𝐷 “ (0[,)𝑑)) ∈ V)
60	59	ralrimivw 3167	. . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑑 ∈ ℝ+ (◡𝐷 “ (0[,)𝑑)) ∈ V)
61		eqid 2769	. . . . . . . . . 10 ⊢ (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) = (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑)))
62		imaeq1 6058	. . . . . . . . . . 11 ⊢ (𝑣 = (◡𝐷 “ (0[,)𝑑)) → (𝑣 “ {𝑥}) = ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}))
63	62	sseq1d 3976	. . . . . . . . . 10 ⊢ (𝑣 = (◡𝐷 “ (0[,)𝑑)) → ((𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
64	61, 63	rexrnmptw 7091	. . . . . . . . 9 ⊢ (∀𝑑 ∈ ℝ+ (◡𝐷 “ (0[,)𝑑)) ∈ V → (∃𝑣 ∈ ran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑)))(𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ∃𝑑 ∈ ℝ+ ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
65	44, 60, 64	3syl 19	. . . . . . . 8 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥 ∈ 𝑎) → (∃𝑣 ∈ ran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑)))(𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ∃𝑑 ∈ ℝ+ ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
66	56, 65	mpbird 260	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥 ∈ 𝑎) → ∃𝑣 ∈ ran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑)))(𝑣 “ {𝑥}) ⊆ 𝑎)
67		oveq2 7419	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑒 → (0[,)𝑑) = (0[,)𝑒))
68	67	imaeq2d 6063	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑒 → (◡𝐷 “ (0[,)𝑑)) = (◡𝐷 “ (0[,)𝑒)))
69	68	cbvmptv 5219	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) = (𝑒 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑒)))
70	69	rneqi 5928	. . . . . . . . . . . 12 ⊢ ran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) = ran (𝑒 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑒)))
71	70	metustfbas 24682	. . . . . . . . . . 11 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) ∈ (fBas‘(𝑋 × 𝑋)))
72		ssfg 23997	. . . . . . . . . . 11 ⊢ (ran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) ∈ (fBas‘(𝑋 × 𝑋)) → ran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) ⊆ ((𝑋 × 𝑋)filGenran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑)))))
73	71, 72	syl 18	. . . . . . . . . 10 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) ⊆ ((𝑋 × 𝑋)filGenran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑)))))
74		metuval 24674	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (metUnif‘𝐷) = ((𝑋 × 𝑋)filGenran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑)))))
75	74	adantl 486	. . . . . . . . . 10 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (metUnif‘𝐷) = ((𝑋 × 𝑋)filGenran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑)))))
76	73, 75	sseqtrrd 3982	. . . . . . . . 9 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) ⊆ (metUnif‘𝐷))
77		ssrexv 4015	. . . . . . . . 9 ⊢ (ran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) ⊆ (metUnif‘𝐷) → (∃𝑣 ∈ ran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑)))(𝑣 “ {𝑥}) ⊆ 𝑎 → ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎))
78	76, 77	syl 18	. . . . . . . 8 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (∃𝑣 ∈ ran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑)))(𝑣 “ {𝑥}) ⊆ 𝑎 → ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎))
79	78	ad2antrr 738	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥 ∈ 𝑎) → (∃𝑣 ∈ ran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑)))(𝑣 “ {𝑥}) ⊆ 𝑎 → ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎))
80	66, 79	mpd 16	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥 ∈ 𝑎) → ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)
81	80	ralrimiva 3163	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)
82	43, 81	jca 520	. . . 4 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎))
83	6	biimpar 482	. . . 4 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)) → 𝑎 ∈ (unifTop‘(metUnif‘𝐷)))
84	82, 83	syldan 602	. . 3 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎 ∈ (unifTop‘(metUnif‘𝐷)))
85	35, 84	impbida 812	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑎 ∈ (unifTop‘(metUnif‘𝐷)) ↔ 𝑎 ∈ (topGen‘ran (ball‘𝐷))))
86	85	eqrdv 2767	1 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (unifTop‘(metUnif‘𝐷)) = (topGen‘ran (ball‘𝐷)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  {crab 3423  Vcvv 3463   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4567  {csn 4594  ∪ cuni 4876   ↦ cmpt 5196   × cxp 5660  ^◡ccnv 5661  ran crn 5663   “ cima 5665  ‘cfv 6537  (class class class)co 7411  0cc0 11099  ℝ⁺crp 13015  [,)cico 13373  topGenctg 17489  PsMetcpsmet 21474  ballcbl 21477  fBascfbas 21478  filGencfg 21479  metUnifcmetu 21481  UnifOncust 24325  unifTopcutop 24355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176  ax-pre-sup 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-er 8693  df-map 8825  df-en 8943  df-dom 8944  df-sdom 8945  df-sup 9401  df-inf 9402  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-div 11871  df-nn 12233  df-2 12302  df-n0 12504  df-z 12591  df-uz 12862  df-q 12972  df-rp 13016  df-xneg 13136  df-xadd 13137  df-xmul 13138  df-ico 13377  df-topgen 17495  df-psmet 21482  df-bl 21485  df-fbas 21487  df-fg 21488  df-metu 21489  df-fil 23971  df-ust 24326  df-utop 24356

This theorem is referenced by:  xmetutop  24693