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Theorem psmetutop 23184
 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.
StepHypRef Expression
1 metuust 23177 . . . . . . . . . . . 12 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (metUnif‘𝐷) ∈ (UnifOn‘𝑋))
2 utopval 22848 . . . . . . . . . . . 12 ((metUnif‘𝐷) ∈ (UnifOn‘𝑋) → (unifTop‘(metUnif‘𝐷)) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎})
31, 2syl 17 . . . . . . . . . . 11 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (unifTop‘(metUnif‘𝐷)) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎})
43eleq2d 2875 . . . . . . . . . 10 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑎 ∈ (unifTop‘(metUnif‘𝐷)) ↔ 𝑎 ∈ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎}))
5 rabid 3331 . . . . . . . . . 10 (𝑎 ∈ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎} ↔ (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎))
64, 5syl6bb 290 . . . . . . . . 9 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑎 ∈ (unifTop‘(metUnif‘𝐷)) ↔ (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)))
76biimpa 480 . . . . . . . 8 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎))
87simpld 498 . . . . . . 7 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → 𝑎 ∈ 𝒫 𝑋)
98elpwid 4508 . . . . . 6 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → 𝑎𝑋)
10 unirnblps 23036 . . . . . . 7 (𝐷 ∈ (PsMet‘𝑋) → ran (ball‘𝐷) = 𝑋)
1110ad2antlr 726 . . . . . 6 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → ran (ball‘𝐷) = 𝑋)
129, 11sseqtrrd 3956 . . . . 5 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → 𝑎 ran (ball‘𝐷))
13 simpr 488 . . . . . . . 8 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → (𝑣 “ {𝑥}) ⊆ 𝑎)
14 simp-5r 785 . . . . . . . . 9 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝐷 ∈ (PsMet‘𝑋))
15 simplr 768 . . . . . . . . 9 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑣 ∈ (metUnif‘𝐷))
169ad3antrrr 729 . . . . . . . . . 10 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑎𝑋)
17 simpllr 775 . . . . . . . . . 10 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑥𝑎)
1816, 17sseldd 3916 . . . . . . . . 9 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑥𝑋)
19 metustbl 23183 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷) ∧ 𝑥𝑋) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏 ⊆ (𝑣 “ {𝑥})))
2014, 15, 18, 19syl3anc 1368 . . . . . . . 8 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏 ⊆ (𝑣 “ {𝑥})))
21 sstr 3923 . . . . . . . . . . 11 ((𝑏 ⊆ (𝑣 “ {𝑥}) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑏𝑎)
2221expcom 417 . . . . . . . . . 10 ((𝑣 “ {𝑥}) ⊆ 𝑎 → (𝑏 ⊆ (𝑣 “ {𝑥}) → 𝑏𝑎))
2322anim2d 614 . . . . . . . . 9 ((𝑣 “ {𝑥}) ⊆ 𝑎 → ((𝑥𝑏𝑏 ⊆ (𝑣 “ {𝑥})) → (𝑥𝑏𝑏𝑎)))
2423reximdv 3232 . . . . . . . 8 ((𝑣 “ {𝑥}) ⊆ 𝑎 → (∃𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏 ⊆ (𝑣 “ {𝑥})) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎)))
2513, 20, 24sylc 65 . . . . . . 7 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎))
267simprd 499 . . . . . . . 8 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)
2726r19.21bi 3173 . . . . . . 7 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) → ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)
2825, 27r19.29a 3248 . . . . . 6 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎))
2928ralrimiva 3149 . . . . 5 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → ∀𝑥𝑎𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎))
3012, 29jca 515 . . . 4 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → (𝑎 ran (ball‘𝐷) ∧ ∀𝑥𝑎𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎)))
31 fvex 6659 . . . . . 6 (ball‘𝐷) ∈ V
3231rnex 7602 . . . . 5 ran (ball‘𝐷) ∈ V
33 eltg2 21573 . . . . 5 (ran (ball‘𝐷) ∈ V → (𝑎 ∈ (topGen‘ran (ball‘𝐷)) ↔ (𝑎 ran (ball‘𝐷) ∧ ∀𝑥𝑎𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎))))
3432, 33mp1i 13 . . . 4 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → (𝑎 ∈ (topGen‘ran (ball‘𝐷)) ↔ (𝑎 ran (ball‘𝐷) ∧ ∀𝑥𝑎𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎))))
3530, 34mpbird 260 . . 3 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → 𝑎 ∈ (topGen‘ran (ball‘𝐷)))
3632, 33mp1i 13 . . . . . . . . 9 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑎 ∈ (topGen‘ran (ball‘𝐷)) ↔ (𝑎 ran (ball‘𝐷) ∧ ∀𝑥𝑎𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎))))
3736biimpa 480 . . . . . . . 8 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → (𝑎 ran (ball‘𝐷) ∧ ∀𝑥𝑎𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎)))
3837simpld 498 . . . . . . 7 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎 ran (ball‘𝐷))
3910ad2antlr 726 . . . . . . 7 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → ran (ball‘𝐷) = 𝑋)
4038, 39sseqtrd 3955 . . . . . 6 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎𝑋)
41 elpwg 4500 . . . . . . 7 (𝑎 ∈ (topGen‘ran (ball‘𝐷)) → (𝑎 ∈ 𝒫 𝑋𝑎𝑋))
4241adantl 485 . . . . . 6 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → (𝑎 ∈ 𝒫 𝑋𝑎𝑋))
4340, 42mpbird 260 . . . . 5 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎 ∈ 𝒫 𝑋)
44 simpllr 775 . . . . . . . . 9 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥𝑎) → 𝐷 ∈ (PsMet‘𝑋))
4540sselda 3915 . . . . . . . . 9 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥𝑎) → 𝑥𝑋)
4637simprd 499 . . . . . . . . . . 11 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → ∀𝑥𝑎𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎))
4746r19.21bi 3173 . . . . . . . . . 10 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥𝑎) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎))
48 blssexps 23043 . . . . . . . . . . 11 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) → (∃𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎) ↔ ∃𝑑 ∈ ℝ+ (𝑥(ball‘𝐷)𝑑) ⊆ 𝑎))
4944, 45, 48syl2anc 587 . . . . . . . . . 10 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥𝑎) → (∃𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎) ↔ ∃𝑑 ∈ ℝ+ (𝑥(ball‘𝐷)𝑑) ⊆ 𝑎))
5047, 49mpbid 235 . . . . . . . . 9 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥𝑎) → ∃𝑑 ∈ ℝ+ (𝑥(ball‘𝐷)𝑑) ⊆ 𝑎)
51 blval2 23179 . . . . . . . . . . . . 13 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋𝑑 ∈ ℝ+) → (𝑥(ball‘𝐷)𝑑) = ((𝐷 “ (0[,)𝑑)) “ {𝑥}))
52513expa 1115 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) ∧ 𝑑 ∈ ℝ+) → (𝑥(ball‘𝐷)𝑑) = ((𝐷 “ (0[,)𝑑)) “ {𝑥}))
5352sseq1d 3946 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) ∧ 𝑑 ∈ ℝ+) → ((𝑥(ball‘𝐷)𝑑) ⊆ 𝑎 ↔ ((𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
5453rexbidva 3255 . . . . . . . . . 10 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) → (∃𝑑 ∈ ℝ+ (𝑥(ball‘𝐷)𝑑) ⊆ 𝑎 ↔ ∃𝑑 ∈ ℝ+ ((𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
5554biimpa 480 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) ∧ ∃𝑑 ∈ ℝ+ (𝑥(ball‘𝐷)𝑑) ⊆ 𝑎) → ∃𝑑 ∈ ℝ+ ((𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎)
5644, 45, 50, 55syl21anc 836 . . . . . . . 8 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥𝑎) → ∃𝑑 ∈ ℝ+ ((𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎)
57 cnvexg 7614 . . . . . . . . . . 11 (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ∈ V)
58 imaexg 7605 . . . . . . . . . . 11 (𝐷 ∈ V → (𝐷 “ (0[,)𝑑)) ∈ V)
5957, 58syl 17 . . . . . . . . . 10 (𝐷 ∈ (PsMet‘𝑋) → (𝐷 “ (0[,)𝑑)) ∈ V)
6059ralrimivw 3150 . . . . . . . . 9 (𝐷 ∈ (PsMet‘𝑋) → ∀𝑑 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) ∈ V)
61 eqid 2798 . . . . . . . . . 10 (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑))) = (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))
62 imaeq1 5892 . . . . . . . . . . 11 (𝑣 = (𝐷 “ (0[,)𝑑)) → (𝑣 “ {𝑥}) = ((𝐷 “ (0[,)𝑑)) “ {𝑥}))
6362sseq1d 3946 . . . . . . . . . 10 (𝑣 = (𝐷 “ (0[,)𝑑)) → ((𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ((𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
6461, 63rexrnmptw 6839 . . . . . . . . 9 (∀𝑑 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) ∈ V → (∃𝑣 ∈ ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))(𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ∃𝑑 ∈ ℝ+ ((𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
6544, 60, 643syl 18 . . . . . . . 8 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥𝑎) → (∃𝑣 ∈ ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))(𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ∃𝑑 ∈ ℝ+ ((𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
6656, 65mpbird 260 . . . . . . 7 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥𝑎) → ∃𝑣 ∈ ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))(𝑣 “ {𝑥}) ⊆ 𝑎)
67 oveq2 7144 . . . . . . . . . . . . . . 15 (𝑑 = 𝑒 → (0[,)𝑑) = (0[,)𝑒))
6867imaeq2d 5897 . . . . . . . . . . . . . 14 (𝑑 = 𝑒 → (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑒)))
6968cbvmptv 5134 . . . . . . . . . . . . 13 (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑))) = (𝑒 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑒)))
7069rneqi 5772 . . . . . . . . . . . 12 ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑))) = ran (𝑒 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑒)))
7170metustfbas 23174 . . . . . . . . . . 11 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑))) ∈ (fBas‘(𝑋 × 𝑋)))
72 ssfg 22487 . . . . . . . . . . 11 (ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑))) ∈ (fBas‘(𝑋 × 𝑋)) → ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑))) ⊆ ((𝑋 × 𝑋)filGenran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))))
7371, 72syl 17 . . . . . . . . . 10 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑))) ⊆ ((𝑋 × 𝑋)filGenran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))))
74 metuval 23166 . . . . . . . . . . 11 (𝐷 ∈ (PsMet‘𝑋) → (metUnif‘𝐷) = ((𝑋 × 𝑋)filGenran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))))
7574adantl 485 . . . . . . . . . 10 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (metUnif‘𝐷) = ((𝑋 × 𝑋)filGenran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))))
7673, 75sseqtrrd 3956 . . . . . . . . 9 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑))) ⊆ (metUnif‘𝐷))
77 ssrexv 3982 . . . . . . . . 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))
7876, 77syl 17 . . . . . . . 8 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (∃𝑣 ∈ ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))(𝑣 “ {𝑥}) ⊆ 𝑎 → ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎))
7978ad2antrr 725 . . . . . . 7 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥𝑎) → (∃𝑣 ∈ ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))(𝑣 “ {𝑥}) ⊆ 𝑎 → ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎))
8066, 79mpd 15 . . . . . 6 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥𝑎) → ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)
8180ralrimiva 3149 . . . . 5 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)
8243, 81jca 515 . . . 4 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎))
836biimpar 481 . . . 4 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)) → 𝑎 ∈ (unifTop‘(metUnif‘𝐷)))
8482, 83syldan 594 . . 3 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎 ∈ (unifTop‘(metUnif‘𝐷)))
8535, 84impbida 800 . 2 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑎 ∈ (unifTop‘(metUnif‘𝐷)) ↔ 𝑎 ∈ (topGen‘ran (ball‘𝐷))))
8685eqrdv 2796 1 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (unifTop‘(metUnif‘𝐷)) = (topGen‘ran (ball‘𝐷)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  {crab 3110  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  𝒫 cpw 4497  {csn 4525  ∪ cuni 4801   ↦ cmpt 5111   × cxp 5518  ◡ccnv 5519  ran crn 5521   “ cima 5523  ‘cfv 6325  (class class class)co 7136  0cc0 10529  ℝ+crp 12380  [,)cico 12731  topGenctg 16706  PsMetcpsmet 20079  ballcbl 20082  fBascfbas 20083  filGencfg 20084  metUnifcmetu 20086  UnifOncust 22815  unifTopcutop 22846 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-1st 7674  df-2nd 7675  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-er 8275  df-map 8394  df-en 8496  df-dom 8497  df-sdom 8498  df-sup 8893  df-inf 8894  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11629  df-2 11691  df-n0 11889  df-z 11973  df-uz 12235  df-q 12340  df-rp 12381  df-xneg 12498  df-xadd 12499  df-xmul 12500  df-ico 12735  df-topgen 16712  df-psmet 20087  df-bl 20090  df-fbas 20092  df-fg 20093  df-metu 20094  df-fil 22461  df-ust 22816  df-utop 22847 This theorem is referenced by:  xmetutop  23185
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