MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  psmetutop Structured version   Visualization version   GIF version

Theorem psmetutop 24005
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 23998 . . . . . . . . . . . 12 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (metUnif‘𝐷) ∈ (UnifOn‘𝑋))
2 utopval 23666 . . . . . . . . . . . 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 2818 . . . . . . . . . 10 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑎 ∈ (unifTop‘(metUnif‘𝐷)) ↔ 𝑎 ∈ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎}))
5 rabid 3451 . . . . . . . . . 10 (𝑎 ∈ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎} ↔ (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎))
64, 5bitrdi 286 . . . . . . . . 9 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑎 ∈ (unifTop‘(metUnif‘𝐷)) ↔ (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)))
76biimpa 477 . . . . . . . 8 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎))
87simpld 495 . . . . . . 7 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → 𝑎 ∈ 𝒫 𝑋)
98elpwid 4605 . . . . . 6 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → 𝑎𝑋)
10 unirnblps 23854 . . . . . . 7 (𝐷 ∈ (PsMet‘𝑋) → ran (ball‘𝐷) = 𝑋)
1110ad2antlr 725 . . . . . 6 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → ran (ball‘𝐷) = 𝑋)
129, 11sseqtrrd 4019 . . . . 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))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → (𝑣 “ {𝑥}) ⊆ 𝑎)
14 simp-5r 784 . . . . . . . . 9 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝐷 ∈ (PsMet‘𝑋))
15 simplr 767 . . . . . . . . 9 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑣 ∈ (metUnif‘𝐷))
169ad3antrrr 728 . . . . . . . . . 10 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑎𝑋)
17 simpllr 774 . . . . . . . . . 10 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑥𝑎)
1816, 17sseldd 3979 . . . . . . . . 9 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑥𝑋)
19 metustbl 24004 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷) ∧ 𝑥𝑋) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏 ⊆ (𝑣 “ {𝑥})))
2014, 15, 18, 19syl3anc 1371 . . . . . . . 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 3986 . . . . . . . . . . 11 ((𝑏 ⊆ (𝑣 “ {𝑥}) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑏𝑎)
2221expcom 414 . . . . . . . . . 10 ((𝑣 “ {𝑥}) ⊆ 𝑎 → (𝑏 ⊆ (𝑣 “ {𝑥}) → 𝑏𝑎))
2322anim2d 612 . . . . . . . . 9 ((𝑣 “ {𝑥}) ⊆ 𝑎 → ((𝑥𝑏𝑏 ⊆ (𝑣 “ {𝑥})) → (𝑥𝑏𝑏𝑎)))
2423reximdv 3169 . . . . . . . 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 496 . . . . . . . 8 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)
2726r19.21bi 3247 . . . . . . 7 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) → ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)
2825, 27r19.29a 3161 . . . . . 6 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎))
2928ralrimiva 3145 . . . . 5 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → ∀𝑥𝑎𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎))
3012, 29jca 512 . . . 4 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → (𝑎 ran (ball‘𝐷) ∧ ∀𝑥𝑎𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎)))
31 fvex 6891 . . . . . 6 (ball‘𝐷) ∈ V
3231rnex 7885 . . . . 5 ran (ball‘𝐷) ∈ V
33 eltg2 22390 . . . . 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 256 . . 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 477 . . . . . . . 8 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → (𝑎 ran (ball‘𝐷) ∧ ∀𝑥𝑎𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎)))
3837simpld 495 . . . . . . 7 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎 ran (ball‘𝐷))
3910ad2antlr 725 . . . . . . 7 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → ran (ball‘𝐷) = 𝑋)
4038, 39sseqtrd 4018 . . . . . 6 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎𝑋)
41 elpwg 4599 . . . . . . 7 (𝑎 ∈ (topGen‘ran (ball‘𝐷)) → (𝑎 ∈ 𝒫 𝑋𝑎𝑋))
4241adantl 482 . . . . . 6 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → (𝑎 ∈ 𝒫 𝑋𝑎𝑋))
4340, 42mpbird 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))) ∧ 𝑥𝑎) → 𝐷 ∈ (PsMet‘𝑋))
4540sselda 3978 . . . . . . . . 9 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥𝑎) → 𝑥𝑋)
4637simprd 496 . . . . . . . . . . 11 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → ∀𝑥𝑎𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎))
4746r19.21bi 3247 . . . . . . . . . 10 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥𝑎) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎))
48 blssexps 23861 . . . . . . . . . . 11 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) → (∃𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎) ↔ ∃𝑑 ∈ ℝ+ (𝑥(ball‘𝐷)𝑑) ⊆ 𝑎))
4944, 45, 48syl2anc 584 . . . . . . . . . 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 231 . . . . . . . . 9 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥𝑎) → ∃𝑑 ∈ ℝ+ (𝑥(ball‘𝐷)𝑑) ⊆ 𝑎)
51 blval2 24000 . . . . . . . . . . . . 13 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋𝑑 ∈ ℝ+) → (𝑥(ball‘𝐷)𝑑) = ((𝐷 “ (0[,)𝑑)) “ {𝑥}))
52513expa 1118 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) ∧ 𝑑 ∈ ℝ+) → (𝑥(ball‘𝐷)𝑑) = ((𝐷 “ (0[,)𝑑)) “ {𝑥}))
5352sseq1d 4009 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) ∧ 𝑑 ∈ ℝ+) → ((𝑥(ball‘𝐷)𝑑) ⊆ 𝑎 ↔ ((𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
5453rexbidva 3175 . . . . . . . . . 10 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) → (∃𝑑 ∈ ℝ+ (𝑥(ball‘𝐷)𝑑) ⊆ 𝑎 ↔ ∃𝑑 ∈ ℝ+ ((𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
5554biimpa 477 . . . . . . . . 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 7897 . . . . . . . . . . 11 (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ∈ V)
58 imaexg 7888 . . . . . . . . . . 11 (𝐷 ∈ V → (𝐷 “ (0[,)𝑑)) ∈ V)
5957, 58syl 17 . . . . . . . . . 10 (𝐷 ∈ (PsMet‘𝑋) → (𝐷 “ (0[,)𝑑)) ∈ V)
6059ralrimivw 3149 . . . . . . . . 9 (𝐷 ∈ (PsMet‘𝑋) → ∀𝑑 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) ∈ V)
61 eqid 2731 . . . . . . . . . 10 (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑))) = (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))
62 imaeq1 6044 . . . . . . . . . . 11 (𝑣 = (𝐷 “ (0[,)𝑑)) → (𝑣 “ {𝑥}) = ((𝐷 “ (0[,)𝑑)) “ {𝑥}))
6362sseq1d 4009 . . . . . . . . . 10 (𝑣 = (𝐷 “ (0[,)𝑑)) → ((𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ((𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
6461, 63rexrnmptw 7081 . . . . . . . . 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 256 . . . . . . 7 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥𝑎) → ∃𝑣 ∈ ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))(𝑣 “ {𝑥}) ⊆ 𝑎)
67 oveq2 7401 . . . . . . . . . . . . . . 15 (𝑑 = 𝑒 → (0[,)𝑑) = (0[,)𝑒))
6867imaeq2d 6049 . . . . . . . . . . . . . 14 (𝑑 = 𝑒 → (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑒)))
6968cbvmptv 5254 . . . . . . . . . . . . 13 (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑))) = (𝑒 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑒)))
7069rneqi 5928 . . . . . . . . . . . 12 ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑))) = ran (𝑒 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑒)))
7170metustfbas 23995 . . . . . . . . . . 11 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑))) ∈ (fBas‘(𝑋 × 𝑋)))
72 ssfg 23305 . . . . . . . . . . 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 23987 . . . . . . . . . . 11 (𝐷 ∈ (PsMet‘𝑋) → (metUnif‘𝐷) = ((𝑋 × 𝑋)filGenran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))))
7574adantl 482 . . . . . . . . . 10 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (metUnif‘𝐷) = ((𝑋 × 𝑋)filGenran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))))
7673, 75sseqtrrd 4019 . . . . . . . . 9 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑))) ⊆ (metUnif‘𝐷))
77 ssrexv 4047 . . . . . . . . 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 724 . . . . . . 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 3145 . . . . 5 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)
8243, 81jca 512 . . . 4 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎))
836biimpar 478 . . . 4 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)) → 𝑎 ∈ (unifTop‘(metUnif‘𝐷)))
8482, 83syldan 591 . . 3 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎 ∈ (unifTop‘(metUnif‘𝐷)))
8535, 84impbida 799 . 2 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑎 ∈ (unifTop‘(metUnif‘𝐷)) ↔ 𝑎 ∈ (topGen‘ran (ball‘𝐷))))
8685eqrdv 2729 1 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (unifTop‘(metUnif‘𝐷)) = (topGen‘ran (ball‘𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2939  wral 3060  wrex 3069  {crab 3431  Vcvv 3473  wss 3944  c0 4318  𝒫 cpw 4596  {csn 4622   cuni 4901  cmpt 5224   × cxp 5667  ccnv 5668  ran crn 5670  cima 5672  cfv 6532  (class class class)co 7393  0cc0 11092  +crp 12956  [,)cico 13308  topGenctg 17365  PsMetcpsmet 20862  ballcbl 20865  fBascfbas 20866  filGencfg 20867  metUnifcmetu 20869  UnifOncust 23633  unifTopcutop 23664
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 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-cnex 11148  ax-resscn 11149  ax-1cn 11150  ax-icn 11151  ax-addcl 11152  ax-addrcl 11153  ax-mulcl 11154  ax-mulrcl 11155  ax-mulcom 11156  ax-addass 11157  ax-mulass 11158  ax-distr 11159  ax-i2m1 11160  ax-1ne0 11161  ax-1rid 11162  ax-rnegex 11163  ax-rrecex 11164  ax-cnre 11165  ax-pre-lttri 11166  ax-pre-lttrn 11167  ax-pre-ltadd 11168  ax-pre-mulgt0 11169  ax-pre-sup 11170
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  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 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-1st 7957  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-er 8686  df-map 8805  df-en 8923  df-dom 8924  df-sdom 8925  df-sup 9419  df-inf 9420  df-pnf 11232  df-mnf 11233  df-xr 11234  df-ltxr 11235  df-le 11236  df-sub 11428  df-neg 11429  df-div 11854  df-nn 12195  df-2 12257  df-n0 12455  df-z 12541  df-uz 12805  df-q 12915  df-rp 12957  df-xneg 13074  df-xadd 13075  df-xmul 13076  df-ico 13312  df-topgen 17371  df-psmet 20870  df-bl 20873  df-fbas 20875  df-fg 20876  df-metu 20877  df-fil 23279  df-ust 23634  df-utop 23665
This theorem is referenced by:  xmetutop  24006
  Copyright terms: Public domain W3C validator