Theorem psmetutop 24777

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 24770	. . . . . . . . . . . 12 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (metUnif‘𝐷) ∈ (UnifOn‘𝑋))
2		utopval 24442	. . . . . . . . . . . 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 2851	. . . . . . . . . 10 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑎 ∈ (unifTop‘(metUnif‘𝐷)) ↔ 𝑎 ∈ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎}))
5		rabid 3439	. . . . . . . . . 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 482	. . . . . . . 8 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎))
8	7	simpld 500	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → 𝑎 ∈ 𝒫 𝑋)
9	8	elpwid 4573	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → 𝑎 ⊆ 𝑋)
10		unirnblps 24629	. . . . . . 7 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∪ ran (ball‘𝐷) = 𝑋)
11	10	ad2antlr 740	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → ∪ ran (ball‘𝐷) = 𝑋)
12	9, 11	sseqtrrd 3975	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → 𝑎 ⊆ ∪ ran (ball‘𝐷))
13		simpr 490	. . . . . . . 8 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → (𝑣 “ {𝑥}) ⊆ 𝑎)
14		simp-5r 798	. . . . . . . . 9 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝐷 ∈ (PsMet‘𝑋))
15		simplr 781	. . . . . . . . 9 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑣 ∈ (metUnif‘𝐷))
16	9	ad3antrrr 743	. . . . . . . . . 10 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑎 ⊆ 𝑋)
17		simpllr 788	. . . . . . . . . 10 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑥 ∈ 𝑎)
18	16, 17	sseldd 3939	. . . . . . . . 9 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑥 ∈ 𝑋)
19		metustbl 24776	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷) ∧ 𝑥 ∈ 𝑋) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ (𝑣 “ {𝑥})))
20	14, 15, 18, 19	syl3anc 1398	. . . . . . . 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 3946	. . . . . . . . . . 11 ⊢ ((𝑏 ⊆ (𝑣 “ {𝑥}) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑏 ⊆ 𝑎)
22	21	expcom 419	. . . . . . . . . 10 ⊢ ((𝑣 “ {𝑥}) ⊆ 𝑎 → (𝑏 ⊆ (𝑣 “ {𝑥}) → 𝑏 ⊆ 𝑎))
23	22	anim2d 624	. . . . . . . . 9 ⊢ ((𝑣 “ {𝑥}) ⊆ 𝑎 → ((𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ (𝑣 “ {𝑥})) → (𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑎)))
24	23	reximdv 3182	. . . . . . . 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 501	. . . . . . . 8 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)
27	26	r19.21bi 3259	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) → ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)
28	25, 27	r19.29a 3175	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑎))
29	28	ralrimiva 3159	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → ∀𝑥 ∈ 𝑎 ∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑎))
30	12, 29	jca 521	. . . 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 7913	. . . . 5 ⊢ ran (ball‘𝐷) ∈ V
33		eltg2 23167	. . . . 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 482	. . . . . . . 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 500	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎 ⊆ ∪ ran (ball‘𝐷))
39	10	ad2antlr 740	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → ∪ ran (ball‘𝐷) = 𝑋)
40	38, 39	sseqtrd 3974	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎 ⊆ 𝑋)
41		elpwg 4567	. . . . . . 7 ⊢ (𝑎 ∈ (topGen‘ran (ball‘𝐷)) → (𝑎 ∈ 𝒫 𝑋 ↔ 𝑎 ⊆ 𝑋))
42	41	adantl 487	. . . . . 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 788	. . . . . . . . 9 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥 ∈ 𝑎) → 𝐷 ∈ (PsMet‘𝑋))
45	40	sselda 3938	. . . . . . . . 9 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥 ∈ 𝑎) → 𝑥 ∈ 𝑋)
46	37	simprd 501	. . . . . . . . . . 11 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → ∀𝑥 ∈ 𝑎 ∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑎))
47	46	r19.21bi 3259	. . . . . . . . . 10 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥 ∈ 𝑎) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑎))
48		blssexps 24636	. . . . . . . . . . 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 596	. . . . . . . . . 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 24772	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑑 ∈ ℝ+) → (𝑥(ball‘𝐷)𝑑) = ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}))
52	51	3expa 1136	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑑 ∈ ℝ+) → (𝑥(ball‘𝐷)𝑑) = ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}))
53	52	sseq1d 3969	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑑 ∈ ℝ+) → ((𝑥(ball‘𝐷)𝑑) ⊆ 𝑎 ↔ ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
54	53	rexbidva 3189	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋) → (∃𝑑 ∈ ℝ+ (𝑥(ball‘𝐷)𝑑) ⊆ 𝑎 ↔ ∃𝑑 ∈ ℝ+ ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
55	54	biimpa 482	. . . . . . . . 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 851	. . . . . . . 8 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥 ∈ 𝑎) → ∃𝑑 ∈ ℝ+ ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎)
57		cnvexg 7927	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ◡𝐷 ∈ V)
58		imaexg 7916	. . . . . . . . . . 11 ⊢ (◡𝐷 ∈ V → (◡𝐷 “ (0[,)𝑑)) ∈ V)
59	57, 58	syl 18	. . . . . . . . . 10 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (◡𝐷 “ (0[,)𝑑)) ∈ V)
60	59	ralrimivw 3163	. . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑑 ∈ ℝ+ (◡𝐷 “ (0[,)𝑑)) ∈ V)
61		eqid 2765	. . . . . . . . . 10 ⊢ (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) = (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑)))
62		imaeq1 6059	. . . . . . . . . . 11 ⊢ (𝑣 = (◡𝐷 “ (0[,)𝑑)) → (𝑣 “ {𝑥}) = ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}))
63	62	sseq1d 3969	. . . . . . . . . 10 ⊢ (𝑣 = (◡𝐷 “ (0[,)𝑑)) → ((𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
64	61, 63	rexrnmptw 7094	. . . . . . . . 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 7427	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑒 → (0[,)𝑑) = (0[,)𝑒))
68	67	imaeq2d 6064	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑒 → (◡𝐷 “ (0[,)𝑑)) = (◡𝐷 “ (0[,)𝑒)))
69	68	cbvmptv 5217	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) = (𝑒 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑒)))
70	69	rneqi 5929	. . . . . . . . . . . 12 ⊢ ran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) = ran (𝑒 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑒)))
71	70	metustfbas 24767	. . . . . . . . . . 11 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) ∈ (fBas‘(𝑋 × 𝑋)))
72		ssfg 24082	. . . . . . . . . . 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 24759	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (metUnif‘𝐷) = ((𝑋 × 𝑋)filGenran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑)))))
75	74	adantl 487	. . . . . . . . . 10 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (metUnif‘𝐷) = ((𝑋 × 𝑋)filGenran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑)))))
76	73, 75	sseqtrrd 3975	. . . . . . . . 9 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) ⊆ (metUnif‘𝐷))
77		ssrexv 4008	. . . . . . . . 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 739	. . . . . . 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 3159	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)
82	43, 81	jca 521	. . . 4 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎))
83	6	biimpar 483	. . . 4 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)) → 𝑎 ∈ (unifTop‘(metUnif‘𝐷)))
84	82, 83	syldan 603	. . 3 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎 ∈ (unifTop‘(metUnif‘𝐷)))
85	35, 84	impbida 813	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑎 ∈ (unifTop‘(metUnif‘𝐷)) ↔ 𝑎 ∈ (topGen‘ran (ball‘𝐷))))
86	85	eqrdv 2763	1 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (unifTop‘(metUnif‘𝐷)) = (topGen‘ran (ball‘𝐷)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  ∃wrex 3091  {crab 3418  Vcvv 3457   ⊆ wss 3906  ∅c0 4286  𝒫 cpw 4564  {csn 4591  ∪ cuni 4874   ↦ cmpt 5194   × cxp 5661  ^◡ccnv 5662  ran crn 5664   “ cima 5666  ‘cfv 6540  (class class class)co 7419  0cc0 11117  ℝ⁺crp 13034  [,)cico 13392  topGenctg 17514  PsMetcpsmet 21558  ballcbl 21561  fBascfbas 21562  filGencfg 21563  metUnifcmetu 21565  UnifOncust 24410  unifTopcutop 24440

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11173  ax-resscn 11174  ax-1cn 11175  ax-icn 11176  ax-addcl 11177  ax-addrcl 11178  ax-mulcl 11179  ax-mulrcl 11180  ax-mulcom 11181  ax-addass 11182  ax-mulass 11183  ax-distr 11184  ax-i2m1 11185  ax-1ne0 11186  ax-1rid 11187  ax-rnegex 11188  ax-rrecex 11189  ax-cnre 11190  ax-pre-lttri 11191  ax-pre-lttrn 11192  ax-pre-ltadd 11193  ax-pre-mulgt0 11194  ax-pre-sup 11195

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-er 8700  df-map 8832  df-en 8950  df-dom 8951  df-sdom 8952  df-sup 9409  df-inf 9410  df-pnf 11262  df-mnf 11263  df-xr 11264  df-ltxr 11265  df-le 11266  df-sub 11460  df-neg 11461  df-div 11889  df-nn 12251  df-2 12320  df-n0 12522  df-z 12609  df-uz 12881  df-q 12991  df-rp 13035  df-xneg 13155  df-xadd 13156  df-xmul 13157  df-ico 13396  df-topgen 17520  df-psmet 21566  df-bl 21569  df-fbas 21571  df-fg 21572  df-metu 21573  df-fil 24056  df-ust 24411  df-utop 24441

This theorem is used by:  xmetutop  24778