Theorem psmetutop 24580

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 24573	. . . . . . . . . . . 12 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (metUnif‘𝐷) ∈ (UnifOn‘𝑋))
2		utopval 24241	. . . . . . . . . . . 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 2827	. . . . . . . . . 10 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑎 ∈ (unifTop‘(metUnif‘𝐷)) ↔ 𝑎 ∈ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎}))
5		rabid 3458	. . . . . . . . . 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 4609	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → 𝑎 ⊆ 𝑋)
10		unirnblps 24429	. . . . . . 7 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∪ ran (ball‘𝐷) = 𝑋)
11	10	ad2antlr 727	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → ∪ ran (ball‘𝐷) = 𝑋)
12	9, 11	sseqtrrd 4021	. . . . 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 786	. . . . . . . . 9 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝐷 ∈ (PsMet‘𝑋))
15		simplr 769	. . . . . . . . 9 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑣 ∈ (metUnif‘𝐷))
16	9	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑎 ⊆ 𝑋)
17		simpllr 776	. . . . . . . . . 10 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑥 ∈ 𝑎)
18	16, 17	sseldd 3984	. . . . . . . . 9 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑥 ∈ 𝑋)
19		metustbl 24579	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷) ∧ 𝑥 ∈ 𝑋) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ (𝑣 “ {𝑥})))
20	14, 15, 18, 19	syl3anc 1373	. . . . . . . 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 3992	. . . . . . . . . . 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 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 495	. . . . . . . 8 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)
27	26	r19.21bi 3251	. . . . . . 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 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 6919	. . . . . 6 ⊢ (ball‘𝐷) ∈ V
32	31	rnex 7932	. . . . 5 ⊢ ran (ball‘𝐷) ∈ V
33		eltg2 22965	. . . . 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 727	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → ∪ ran (ball‘𝐷) = 𝑋)
40	38, 39	sseqtrd 4020	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎 ⊆ 𝑋)
41		elpwg 4603	. . . . . . 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 776	. . . . . . . . 9 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥 ∈ 𝑎) → 𝐷 ∈ (PsMet‘𝑋))
45	40	sselda 3983	. . . . . . . . 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 3251	. . . . . . . . . 10 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥 ∈ 𝑎) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑏 ∧ 𝑏 ⊆ 𝑎))
48		blssexps 24436	. . . . . . . . . . 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 232	. . . . . . . . 9 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥 ∈ 𝑎) → ∃𝑑 ∈ ℝ+ (𝑥(ball‘𝐷)𝑑) ⊆ 𝑎)
51		blval2 24575	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑑 ∈ ℝ+) → (𝑥(ball‘𝐷)𝑑) = ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}))
52	51	3expa 1119	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑑 ∈ ℝ+) → (𝑥(ball‘𝐷)𝑑) = ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}))
53	52	sseq1d 4015	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑑 ∈ ℝ+) → ((𝑥(ball‘𝐷)𝑑) ⊆ 𝑎 ↔ ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
54	53	rexbidva 3177	. . . . . . . . . 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 838	. . . . . . . 8 ⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥 ∈ 𝑎) → ∃𝑑 ∈ ℝ+ ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎)
57		cnvexg 7946	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ◡𝐷 ∈ V)
58		imaexg 7935	. . . . . . . . . . 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 2737	. . . . . . . . . 10 ⊢ (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) = (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑)))
62		imaeq1 6073	. . . . . . . . . . 11 ⊢ (𝑣 = (◡𝐷 “ (0[,)𝑑)) → (𝑣 “ {𝑥}) = ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}))
63	62	sseq1d 4015	. . . . . . . . . 10 ⊢ (𝑣 = (◡𝐷 “ (0[,)𝑑)) → ((𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ((◡𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
64	61, 63	rexrnmptw 7115	. . . . . . . . 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 7439	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑒 → (0[,)𝑑) = (0[,)𝑒))
68	67	imaeq2d 6078	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑒 → (◡𝐷 “ (0[,)𝑑)) = (◡𝐷 “ (0[,)𝑒)))
69	68	cbvmptv 5255	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) = (𝑒 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑒)))
70	69	rneqi 5948	. . . . . . . . . . . 12 ⊢ ran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) = ran (𝑒 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑒)))
71	70	metustfbas 24570	. . . . . . . . . . 11 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) ∈ (fBas‘(𝑋 × 𝑋)))
72		ssfg 23880	. . . . . . . . . . 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 24562	. . . . . . . . . . 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 4021	. . . . . . . . 9 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) ⊆ (metUnif‘𝐷))
77		ssrexv 4053	. . . . . . . . 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 726	. . . . . . 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 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 591	. . 3 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎 ∈ (unifTop‘(metUnif‘𝐷)))
85	35, 84	impbida 801	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑎 ∈ (unifTop‘(metUnif‘𝐷)) ↔ 𝑎 ∈ (topGen‘ran (ball‘𝐷))))
86	85	eqrdv 2735	1 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (unifTop‘(metUnif‘𝐷)) = (topGen‘ran (ball‘𝐷)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  {crab 3436  Vcvv 3480   ⊆ wss 3951  ∅c0 4333  𝒫 cpw 4600  {csn 4626  ∪ cuni 4907   ↦ cmpt 5225   × cxp 5683  ^◡ccnv 5684  ran crn 5686   “ cima 5688  ‘cfv 6561  (class class class)co 7431  0cc0 11155  ℝ⁺crp 13034  [,)cico 13389  topGenctg 17482  PsMetcpsmet 21348  ballcbl 21351  fBascfbas 21352  filGencfg 21353  metUnifcmetu 21355  UnifOncust 24208  unifTopcutop 24239

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ico 13393  df-topgen 17488  df-psmet 21356  df-bl 21359  df-fbas 21361  df-fg 21362  df-metu 21363  df-fil 23854  df-ust 24209  df-utop 24240

This theorem is referenced by:  xmetutop  24581