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Theorem psmetutop 22865
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 22858 . . . . . . . . . . . 12 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (metUnif‘𝐷) ∈ (UnifOn‘𝑋))
2 utopval 22529 . . . . . . . . . . . 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 2868 . . . . . . . . . 10 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑎 ∈ (unifTop‘(metUnif‘𝐷)) ↔ 𝑎 ∈ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎}))
5 rabid 3337 . . . . . . . . . 10 (𝑎 ∈ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎} ↔ (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎))
64, 5syl6bb 288 . . . . . . . . 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 4469 . . . . . 6 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → 𝑎𝑋)
10 unirnblps 22717 . . . . . . 7 (𝐷 ∈ (PsMet‘𝑋) → ran (ball‘𝐷) = 𝑋)
1110ad2antlr 723 . . . . . 6 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → ran (ball‘𝐷) = 𝑋)
129, 11sseqtr4d 3933 . . . . 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 782 . . . . . . . . 9 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝐷 ∈ (PsMet‘𝑋))
15 simplr 765 . . . . . . . . 9 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑣 ∈ (metUnif‘𝐷))
169ad3antrrr 726 . . . . . . . . . 10 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑎𝑋)
17 simpllr 772 . . . . . . . . . 10 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑥𝑎)
1816, 17sseldd 3894 . . . . . . . . 9 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑥𝑋)
19 metustbl 22864 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷) ∧ 𝑥𝑋) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏 ⊆ (𝑣 “ {𝑥})))
2014, 15, 18, 19syl3anc 1364 . . . . . . . 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 3901 . . . . . . . . . . 11 ((𝑏 ⊆ (𝑣 “ {𝑥}) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑏𝑎)
2221expcom 414 . . . . . . . . . 10 ((𝑣 “ {𝑥}) ⊆ 𝑎 → (𝑏 ⊆ (𝑣 “ {𝑥}) → 𝑏𝑎))
2322anim2d 611 . . . . . . . . 9 ((𝑣 “ {𝑥}) ⊆ 𝑎 → ((𝑥𝑏𝑏 ⊆ (𝑣 “ {𝑥})) → (𝑥𝑏𝑏𝑎)))
2423reximdv 3236 . . . . . . . 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 3175 . . . . . . 7 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) → ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)
2825, 27r19.29a 3252 . . . . . 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 512 . . . 4 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → (𝑎 ran (ball‘𝐷) ∧ ∀𝑥𝑎𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎)))
31 fvex 6556 . . . . . 6 (ball‘𝐷) ∈ V
3231rnex 7478 . . . . 5 ran (ball‘𝐷) ∈ V
33 eltg2 21255 . . . . 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 258 . . 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 723 . . . . . . 7 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → ran (ball‘𝐷) = 𝑋)
4038, 39sseqtrd 3932 . . . . . 6 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎𝑋)
41 elpwg 4465 . . . . . . 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 258 . . . . 5 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎 ∈ 𝒫 𝑋)
44 simpllr 772 . . . . . . . . 9 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥𝑎) → 𝐷 ∈ (PsMet‘𝑋))
4540sselda 3893 . . . . . . . . 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 3175 . . . . . . . . . 10 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥𝑎) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎))
48 blssexps 22724 . . . . . . . . . . 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 233 . . . . . . . . 9 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥𝑎) → ∃𝑑 ∈ ℝ+ (𝑥(ball‘𝐷)𝑑) ⊆ 𝑎)
51 blval2 22860 . . . . . . . . . . . . 13 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋𝑑 ∈ ℝ+) → (𝑥(ball‘𝐷)𝑑) = ((𝐷 “ (0[,)𝑑)) “ {𝑥}))
52513expa 1111 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) ∧ 𝑑 ∈ ℝ+) → (𝑥(ball‘𝐷)𝑑) = ((𝐷 “ (0[,)𝑑)) “ {𝑥}))
5352sseq1d 3923 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) ∧ 𝑑 ∈ ℝ+) → ((𝑥(ball‘𝐷)𝑑) ⊆ 𝑎 ↔ ((𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
5453rexbidva 3259 . . . . . . . . . 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 834 . . . . . . . 8 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥𝑎) → ∃𝑑 ∈ ℝ+ ((𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎)
57 cnvexg 7490 . . . . . . . . . . 11 (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ∈ V)
58 imaexg 7481 . . . . . . . . . . 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 2795 . . . . . . . . . 10 (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑))) = (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))
62 imaeq1 5806 . . . . . . . . . . 11 (𝑣 = (𝐷 “ (0[,)𝑑)) → (𝑣 “ {𝑥}) = ((𝐷 “ (0[,)𝑑)) “ {𝑥}))
6362sseq1d 3923 . . . . . . . . . 10 (𝑣 = (𝐷 “ (0[,)𝑑)) → ((𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ((𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
6461, 63rexrnmpt 6731 . . . . . . . . 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 258 . . . . . . 7 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥𝑎) → ∃𝑣 ∈ ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))(𝑣 “ {𝑥}) ⊆ 𝑎)
67 oveq2 7029 . . . . . . . . . . . . . . 15 (𝑑 = 𝑒 → (0[,)𝑑) = (0[,)𝑒))
6867imaeq2d 5811 . . . . . . . . . . . . . 14 (𝑑 = 𝑒 → (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑒)))
6968cbvmptv 5066 . . . . . . . . . . . . 13 (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑))) = (𝑒 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑒)))
7069rneqi 5694 . . . . . . . . . . . 12 ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑))) = ran (𝑒 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑒)))
7170metustfbas 22855 . . . . . . . . . . 11 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑))) ∈ (fBas‘(𝑋 × 𝑋)))
72 ssfg 22169 . . . . . . . . . . 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 22847 . . . . . . . . . . 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, 75sseqtr4d 3933 . . . . . . . . 9 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑))) ⊆ (metUnif‘𝐷))
77 ssrexv 3959 . . . . . . . . 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 722 . . . . . . 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 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 797 . 2 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑎 ∈ (unifTop‘(metUnif‘𝐷)) ↔ 𝑎 ∈ (topGen‘ran (ball‘𝐷))))
8685eqrdv 2793 1 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (unifTop‘(metUnif‘𝐷)) = (topGen‘ran (ball‘𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wcel 2081  wne 2984  wral 3105  wrex 3106  {crab 3109  Vcvv 3437  wss 3863  c0 4215  𝒫 cpw 4457  {csn 4476   cuni 4749  cmpt 5045   × cxp 5446  ccnv 5447  ran crn 5449  cima 5451  cfv 6230  (class class class)co 7021  0cc0 10388  +crp 12244  [,)cico 12595  topGenctg 16545  PsMetcpsmet 20216  ballcbl 20219  fBascfbas 20220  filGencfg 20221  metUnifcmetu 20223  UnifOncust 22496  unifTopcutop 22527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324  ax-cnex 10444  ax-resscn 10445  ax-1cn 10446  ax-icn 10447  ax-addcl 10448  ax-addrcl 10449  ax-mulcl 10450  ax-mulrcl 10451  ax-mulcom 10452  ax-addass 10453  ax-mulass 10454  ax-distr 10455  ax-i2m1 10456  ax-1ne0 10457  ax-1rid 10458  ax-rnegex 10459  ax-rrecex 10460  ax-cnre 10461  ax-pre-lttri 10462  ax-pre-lttrn 10463  ax-pre-ltadd 10464  ax-pre-mulgt0 10465  ax-pre-sup 10466
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-pss 3880  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-tp 4481  df-op 4483  df-uni 4750  df-iun 4831  df-br 4967  df-opab 5029  df-mpt 5046  df-tr 5069  df-id 5353  df-eprel 5358  df-po 5367  df-so 5368  df-fr 5407  df-we 5409  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-pred 6028  df-ord 6074  df-on 6075  df-lim 6076  df-suc 6077  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-riota 6982  df-ov 7024  df-oprab 7025  df-mpo 7026  df-om 7442  df-1st 7550  df-2nd 7551  df-wrecs 7803  df-recs 7865  df-rdg 7903  df-er 8144  df-map 8263  df-en 8363  df-dom 8364  df-sdom 8365  df-sup 8757  df-inf 8758  df-pnf 10528  df-mnf 10529  df-xr 10530  df-ltxr 10531  df-le 10532  df-sub 10724  df-neg 10725  df-div 11151  df-nn 11492  df-2 11553  df-n0 11751  df-z 11835  df-uz 12099  df-q 12203  df-rp 12245  df-xneg 12362  df-xadd 12363  df-xmul 12364  df-ico 12599  df-topgen 16551  df-psmet 20224  df-bl 20227  df-fbas 20229  df-fg 20230  df-metu 20231  df-fil 22143  df-ust 22497  df-utop 22528
This theorem is referenced by:  xmetutop  22866
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