Theorem psmetutop 24075

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 24068	. . . . . . . . . . . 12 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (metUnif‘𝐷) ∈ (UnifOn‘𝑋))
2		utopval 23736	. . . . . . . . . . . 12 ⊢ ((metUnif‘𝐷) ∈ (UnifOn‘𝑋) → (unifTop‘(metUnif‘𝐷)) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎})
3	1, 2	syl 17	. . . . . . . . . . 11 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (unifTop‘(metUnif‘𝐷)) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎})
4	3	eleq2d 2819	. . . . . . . . . 10 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑎 ∈ (unifTop‘(metUnif‘𝐷)) ↔ 𝑎 ∈ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎}))
5		rabid 3452	. . . . . . . . . 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 286	. . . . . . . . 9 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑎 ∈ (unifTop‘(metUnif‘𝐷)) ↔ (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)))
7	6	biimpa 477	. . . . . . . 8 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎))
8	7	simpld 495	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → 𝑎 ∈ 𝒫 𝑋)
9	8	elpwid 4611	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → 𝑎 ⊆ 𝑋)
10		unirnblps 23924	. . . . . . 7 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∪ ran (ball‘𝐷) = 𝑋)
11	10	ad2antlr 725	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → ∪ ran (ball‘𝐷) = 𝑋)
12	9, 11	sseqtrrd 4023	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → 𝑎 ⊆ ∪ ran (ball‘𝐷))
13		simpr 485	. . . . . . . 8 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → (𝑣 “ {𝑥}) ⊆ 𝑎)
14		simp-5r 784	. . . . . . . . 9 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝐷 ∈ (PsMet‘𝑋))
15		simplr 767	. . . . . . . . 9 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑣 ∈ (metUnif‘𝐷))
16	9	ad3antrrr 728	. . . . . . . . . 10 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑎 ⊆ 𝑋)
17		simpllr 774	. . . . . . . . . 10 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑥 ∈ 𝑎)
18	16, 17	sseldd 3983	. . . . . . . . 9 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑥 ∈ 𝑋)
19		metustbl 24074	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷) ∧ 𝑥 ∈ 𝑋) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ (𝑣 “ {𝑥})))
20	14, 15, 18, 19	syl3anc 1371	. . . . . . . 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 3990	. . . . . . . . . . 11 ⊢ ((𝑏 ⊆ (𝑣 “ {𝑥}) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑏 ⊆ 𝑎)
22	21	expcom 414	. . . . . . . . . 10 ⊢ ((𝑣 “ {𝑥}) ⊆ 𝑎 → (𝑏 ⊆ (𝑣 “ {𝑥}) → 𝑏 ⊆ 𝑎))
23	22	anim2d 612	. . . . . . . . 9 ⊢ ((𝑣 “ {𝑥}) ⊆ 𝑎 → ((𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ (𝑣 “ {𝑥})) → (𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑎)))
24	23	reximdv 3170	. . . . . . . 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 65	. . . . . . 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 496	. . . . . . . 8 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)
27	26	r19.21bi 3248	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) → ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)
28	25, 27	r19.29a 3162	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑎))
29	28	ralrimiva 3146	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → ∀𝑥 ∈ 𝑎 ∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑎))
30	12, 29	jca 512	. . . 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 6904	. . . . . 6 ⊢ (ball‘𝐷) ∈ V
32	31	rnex 7902	. . . . 5 ⊢ ran (ball‘𝐷) ∈ V
33		eltg2 22460	. . . . 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 13	. . . 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 256	. . 3 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → 𝑎 ∈ (topGen‘ran (ball‘𝐷)))
36	32, 33	mp1i 13	. . . . . . . . 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 477	. . . . . . . 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 495	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎 ⊆ ∪ ran (ball‘𝐷))
39	10	ad2antlr 725	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → ∪ ran (ball‘𝐷) = 𝑋)
40	38, 39	sseqtrd 4022	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎 ⊆ 𝑋)
41		elpwg 4605	. . . . . . 7 ⊢ (𝑎 ∈ (topGen‘ran (ball‘𝐷)) → (𝑎 ∈ 𝒫 𝑋 ↔ 𝑎 ⊆ 𝑋))
42	41	adantl 482	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → (𝑎 ∈ 𝒫 𝑋 ↔ 𝑎 ⊆ 𝑋))
43	40, 42	mpbird 256	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎 ∈ 𝒫 𝑋)
44		simpllr 774	. . . . . . . . 9 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥 ∈ 𝑎) → 𝐷 ∈ (PsMet‘𝑋))
45	40	sselda 3982	. . . . . . . . 9 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥 ∈ 𝑎) → 𝑥 ∈ 𝑋)
46	37	simprd 496	. . . . . . . . . . 11 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → ∀𝑥 ∈ 𝑎 ∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑎))
47	46	r19.21bi 3248	. . . . . . . . . 10 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥 ∈ 𝑎) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑎))
48		blssexps 23931	. . . . . . . . . . 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 584	. . . . . . . . . 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 231	. . . . . . . . 9 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥 ∈ 𝑎) → ∃𝑑 ∈ ℝ+ (𝑥(ball‘𝐷)𝑑) ⊆ 𝑎)
51		blval2 24070	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑑 ∈ ℝ+) → (𝑥(ball‘𝐷)𝑑) = ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}))
52	51	3expa 1118	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑑 ∈ ℝ+) → (𝑥(ball‘𝐷)𝑑) = ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}))
53	52	sseq1d 4013	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑑 ∈ ℝ+) → ((𝑥(ball‘𝐷)𝑑) ⊆ 𝑎 ↔ ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
54	53	rexbidva 3176	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋) → (∃𝑑 ∈ ℝ+ (𝑥(ball‘𝐷)𝑑) ⊆ 𝑎 ↔ ∃𝑑 ∈ ℝ+ ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
55	54	biimpa 477	. . . . . . . . 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 836	. . . . . . . 8 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥 ∈ 𝑎) → ∃𝑑 ∈ ℝ+ ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎)
57		cnvexg 7914	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ◡𝐷 ∈ V)
58		imaexg 7905	. . . . . . . . . . 11 ⊢ (◡𝐷 ∈ V → (◡𝐷 “ (0[,)𝑑)) ∈ V)
59	57, 58	syl 17	. . . . . . . . . 10 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (◡𝐷 “ (0[,)𝑑)) ∈ V)
60	59	ralrimivw 3150	. . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑑 ∈ ℝ+ (◡𝐷 “ (0[,)𝑑)) ∈ V)
61		eqid 2732	. . . . . . . . . 10 ⊢ (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) = (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑)))
62		imaeq1 6054	. . . . . . . . . . 11 ⊢ (𝑣 = (◡𝐷 “ (0[,)𝑑)) → (𝑣 “ {𝑥}) = ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}))
63	62	sseq1d 4013	. . . . . . . . . 10 ⊢ (𝑣 = (◡𝐷 “ (0[,)𝑑)) → ((𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
64	61, 63	rexrnmptw 7096	. . . . . . . . 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 18	. . . . . . . 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 256	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥 ∈ 𝑎) → ∃𝑣 ∈ ran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑)))(𝑣 “ {𝑥}) ⊆ 𝑎)
67		oveq2 7416	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑒 → (0[,)𝑑) = (0[,)𝑒))
68	67	imaeq2d 6059	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑒 → (◡𝐷 “ (0[,)𝑑)) = (◡𝐷 “ (0[,)𝑒)))
69	68	cbvmptv 5261	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) = (𝑒 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑒)))
70	69	rneqi 5936	. . . . . . . . . . . 12 ⊢ ran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) = ran (𝑒 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑒)))
71	70	metustfbas 24065	. . . . . . . . . . 11 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) ∈ (fBas‘(𝑋 × 𝑋)))
72		ssfg 23375	. . . . . . . . . . 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 17	. . . . . . . . . 10 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) ⊆ ((𝑋 × 𝑋)filGenran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑)))))
74		metuval 24057	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (metUnif‘𝐷) = ((𝑋 × 𝑋)filGenran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑)))))
75	74	adantl 482	. . . . . . . . . 10 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (metUnif‘𝐷) = ((𝑋 × 𝑋)filGenran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑)))))
76	73, 75	sseqtrrd 4023	. . . . . . . . 9 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) ⊆ (metUnif‘𝐷))
77		ssrexv 4051	. . . . . . . . 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 17	. . . . . . . 8 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (∃𝑣 ∈ ran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑)))(𝑣 “ {𝑥}) ⊆ 𝑎 → ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎))
79	78	ad2antrr 724	. . . . . . 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 15	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥 ∈ 𝑎) → ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)
81	80	ralrimiva 3146	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)
82	43, 81	jca 512	. . . 4 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎))
83	6	biimpar 478	. . . 4 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)) → 𝑎 ∈ (unifTop‘(metUnif‘𝐷)))
84	82, 83	syldan 591	. . 3 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎 ∈ (unifTop‘(metUnif‘𝐷)))
85	35, 84	impbida 799	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑎 ∈ (unifTop‘(metUnif‘𝐷)) ↔ 𝑎 ∈ (topGen‘ran (ball‘𝐷))))
86	85	eqrdv 2730	1 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (unifTop‘(metUnif‘𝐷)) = (topGen‘ran (ball‘𝐷)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  {crab 3432  Vcvv 3474   ⊆ wss 3948  ∅c0 4322  𝒫 cpw 4602  {csn 4628  ∪ cuni 4908   ↦ cmpt 5231   × cxp 5674  ^◡ccnv 5675  ran crn 5677   “ cima 5679  ‘cfv 6543  (class class class)co 7408  0cc0 11109  ℝ⁺crp 12973  [,)cico 13325  topGenctg 17382  PsMetcpsmet 20927  ballcbl 20930  fBascfbas 20931  filGencfg 20932  metUnifcmetu 20934  UnifOncust 23703  unifTopcutop 23734

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  ax-pre-sup 11187

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-pss 3967  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-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  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-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  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-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-sup 9436  df-inf 9437  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-nn 12212  df-2 12274  df-n0 12472  df-z 12558  df-uz 12822  df-q 12932  df-rp 12974  df-xneg 13091  df-xadd 13092  df-xmul 13093  df-ico 13329  df-topgen 17388  df-psmet 20935  df-bl 20938  df-fbas 20940  df-fg 20941  df-metu 20942  df-fil 23349  df-ust 23704  df-utop 23735

This theorem is referenced by:  xmetutop  24076