Theorem psmetutop 24431

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 24424	. . . . . . . . . . . 12 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (metUnif‘𝐷) ∈ (UnifOn‘𝑋))
2		utopval 24092	. . . . . . . . . . . 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 2813	. . . . . . . . . 10 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑎 ∈ (unifTop‘(metUnif‘𝐷)) ↔ 𝑎 ∈ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎}))
5		rabid 3446	. . . . . . . . . 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 287	. . . . . . . . 9 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑎 ∈ (unifTop‘(metUnif‘𝐷)) ↔ (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)))
7	6	biimpa 476	. . . . . . . 8 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎))
8	7	simpld 494	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → 𝑎 ∈ 𝒫 𝑋)
9	8	elpwid 4606	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → 𝑎 ⊆ 𝑋)
10		unirnblps 24280	. . . . . . 7 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∪ ran (ball‘𝐷) = 𝑋)
11	10	ad2antlr 724	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → ∪ ran (ball‘𝐷) = 𝑋)
12	9, 11	sseqtrrd 4018	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → 𝑎 ⊆ ∪ ran (ball‘𝐷))
13		simpr 484	. . . . . . . 8 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → (𝑣 “ {𝑥}) ⊆ 𝑎)
14		simp-5r 783	. . . . . . . . 9 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝐷 ∈ (PsMet‘𝑋))
15		simplr 766	. . . . . . . . 9 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑣 ∈ (metUnif‘𝐷))
16	9	ad3antrrr 727	. . . . . . . . . 10 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑎 ⊆ 𝑋)
17		simpllr 773	. . . . . . . . . 10 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑥 ∈ 𝑎)
18	16, 17	sseldd 3978	. . . . . . . . 9 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑥 ∈ 𝑋)
19		metustbl 24430	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷) ∧ 𝑥 ∈ 𝑋) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ (𝑣 “ {𝑥})))
20	14, 15, 18, 19	syl3anc 1368	. . . . . . . 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 3985	. . . . . . . . . . 11 ⊢ ((𝑏 ⊆ (𝑣 “ {𝑥}) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑏 ⊆ 𝑎)
22	21	expcom 413	. . . . . . . . . 10 ⊢ ((𝑣 “ {𝑥}) ⊆ 𝑎 → (𝑏 ⊆ (𝑣 “ {𝑥}) → 𝑏 ⊆ 𝑎))
23	22	anim2d 611	. . . . . . . . 9 ⊢ ((𝑣 “ {𝑥}) ⊆ 𝑎 → ((𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ (𝑣 “ {𝑥})) → (𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑎)))
24	23	reximdv 3164	. . . . . . . 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 495	. . . . . . . 8 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)
27	26	r19.21bi 3242	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) → ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)
28	25, 27	r19.29a 3156	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑎))
29	28	ralrimiva 3140	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → ∀𝑥 ∈ 𝑎 ∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑎))
30	12, 29	jca 511	. . . 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 6898	. . . . . 6 ⊢ (ball‘𝐷) ∈ V
32	31	rnex 7900	. . . . 5 ⊢ ran (ball‘𝐷) ∈ V
33		eltg2 22816	. . . . 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 257	. . 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 476	. . . . . . . 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 494	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎 ⊆ ∪ ran (ball‘𝐷))
39	10	ad2antlr 724	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → ∪ ran (ball‘𝐷) = 𝑋)
40	38, 39	sseqtrd 4017	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎 ⊆ 𝑋)
41		elpwg 4600	. . . . . . 7 ⊢ (𝑎 ∈ (topGen‘ran (ball‘𝐷)) → (𝑎 ∈ 𝒫 𝑋 ↔ 𝑎 ⊆ 𝑋))
42	41	adantl 481	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → (𝑎 ∈ 𝒫 𝑋 ↔ 𝑎 ⊆ 𝑋))
43	40, 42	mpbird 257	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎 ∈ 𝒫 𝑋)
44		simpllr 773	. . . . . . . . 9 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥 ∈ 𝑎) → 𝐷 ∈ (PsMet‘𝑋))
45	40	sselda 3977	. . . . . . . . 9 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥 ∈ 𝑎) → 𝑥 ∈ 𝑋)
46	37	simprd 495	. . . . . . . . . . 11 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → ∀𝑥 ∈ 𝑎 ∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑎))
47	46	r19.21bi 3242	. . . . . . . . . 10 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥 ∈ 𝑎) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑎))
48		blssexps 24287	. . . . . . . . . . 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 583	. . . . . . . . . 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 24426	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑑 ∈ ℝ+) → (𝑥(ball‘𝐷)𝑑) = ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}))
52	51	3expa 1115	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑑 ∈ ℝ+) → (𝑥(ball‘𝐷)𝑑) = ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}))
53	52	sseq1d 4008	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑑 ∈ ℝ+) → ((𝑥(ball‘𝐷)𝑑) ⊆ 𝑎 ↔ ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
54	53	rexbidva 3170	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋) → (∃𝑑 ∈ ℝ+ (𝑥(ball‘𝐷)𝑑) ⊆ 𝑎 ↔ ∃𝑑 ∈ ℝ+ ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
55	54	biimpa 476	. . . . . . . . 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 835	. . . . . . . 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 7903	. . . . . . . . . . 11 ⊢ (◡𝐷 ∈ V → (◡𝐷 “ (0[,)𝑑)) ∈ V)
59	57, 58	syl 17	. . . . . . . . . 10 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (◡𝐷 “ (0[,)𝑑)) ∈ V)
60	59	ralrimivw 3144	. . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑑 ∈ ℝ+ (◡𝐷 “ (0[,)𝑑)) ∈ V)
61		eqid 2726	. . . . . . . . . 10 ⊢ (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) = (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑)))
62		imaeq1 6048	. . . . . . . . . . 11 ⊢ (𝑣 = (◡𝐷 “ (0[,)𝑑)) → (𝑣 “ {𝑥}) = ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}))
63	62	sseq1d 4008	. . . . . . . . . 10 ⊢ (𝑣 = (◡𝐷 “ (0[,)𝑑)) → ((𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
64	61, 63	rexrnmptw 7090	. . . . . . . . 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 257	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥 ∈ 𝑎) → ∃𝑣 ∈ ran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑)))(𝑣 “ {𝑥}) ⊆ 𝑎)
67		oveq2 7413	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑒 → (0[,)𝑑) = (0[,)𝑒))
68	67	imaeq2d 6053	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑒 → (◡𝐷 “ (0[,)𝑑)) = (◡𝐷 “ (0[,)𝑒)))
69	68	cbvmptv 5254	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) = (𝑒 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑒)))
70	69	rneqi 5930	. . . . . . . . . . . 12 ⊢ ran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) = ran (𝑒 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑒)))
71	70	metustfbas 24421	. . . . . . . . . . 11 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) ∈ (fBas‘(𝑋 × 𝑋)))
72		ssfg 23731	. . . . . . . . . . 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 24413	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (metUnif‘𝐷) = ((𝑋 × 𝑋)filGenran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑)))))
75	74	adantl 481	. . . . . . . . . 10 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (metUnif‘𝐷) = ((𝑋 × 𝑋)filGenran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑)))))
76	73, 75	sseqtrrd 4018	. . . . . . . . 9 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) ⊆ (metUnif‘𝐷))
77		ssrexv 4046	. . . . . . . . 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 723	. . . . . . 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 3140	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)
82	43, 81	jca 511	. . . 4 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎))
83	6	biimpar 477	. . . 4 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)) → 𝑎 ∈ (unifTop‘(metUnif‘𝐷)))
84	82, 83	syldan 590	. . 3 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎 ∈ (unifTop‘(metUnif‘𝐷)))
85	35, 84	impbida 798	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑎 ∈ (unifTop‘(metUnif‘𝐷)) ↔ 𝑎 ∈ (topGen‘ran (ball‘𝐷))))
86	85	eqrdv 2724	1 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (unifTop‘(metUnif‘𝐷)) = (topGen‘ran (ball‘𝐷)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098   ≠ wne 2934  ∀wral 3055  ∃wrex 3064  {crab 3426  Vcvv 3468   ⊆ wss 3943  ∅c0 4317  𝒫 cpw 4597  {csn 4623  ∪ cuni 4902   ↦ cmpt 5224   × cxp 5667  ^◡ccnv 5668  ran crn 5670   “ cima 5672  ‘cfv 6537  (class class class)co 7405  0cc0 11112  ℝ⁺crp 12980  [,)cico 13332  topGenctg 17392  PsMetcpsmet 21224  ballcbl 21227  fBascfbas 21228  filGencfg 21229  metUnifcmetu 21231  UnifOncust 24059  unifTopcutop 24090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-sup 9439  df-inf 9440  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-n0 12477  df-z 12563  df-uz 12827  df-q 12937  df-rp 12981  df-xneg 13098  df-xadd 13099  df-xmul 13100  df-ico 13336  df-topgen 17398  df-psmet 21232  df-bl 21235  df-fbas 21237  df-fg 21238  df-metu 21239  df-fil 23705  df-ust 24060  df-utop 24091

This theorem is referenced by:  xmetutop  24432