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Theorem psmetutop 22665
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 22658 . . . . . . . . . . . 12 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (metUnif‘𝐷) ∈ (UnifOn‘𝑋))
2 utopval 22329 . . . . . . . . . . . 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 2830 . . . . . . . . . 10 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑎 ∈ (unifTop‘(metUnif‘𝐷)) ↔ 𝑎 ∈ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎}))
5 rabid 3263 . . . . . . . . . 10 (𝑎 ∈ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎} ↔ (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎))
64, 5syl6bb 278 . . . . . . . . 9 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑎 ∈ (unifTop‘(metUnif‘𝐷)) ↔ (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)))
76biimpa 468 . . . . . . . 8 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎))
87simpld 488 . . . . . . 7 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → 𝑎 ∈ 𝒫 𝑋)
98elpwid 4329 . . . . . 6 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → 𝑎𝑋)
10 unirnblps 22517 . . . . . . 7 (𝐷 ∈ (PsMet‘𝑋) → ran (ball‘𝐷) = 𝑋)
1110ad2antlr 718 . . . . . 6 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → ran (ball‘𝐷) = 𝑋)
129, 11sseqtr4d 3804 . . . . 5 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → 𝑎 ran (ball‘𝐷))
13 simpr 477 . . . . . . . 8 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → (𝑣 “ {𝑥}) ⊆ 𝑎)
14 simp-5r 807 . . . . . . . . 9 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝐷 ∈ (PsMet‘𝑋))
15 simplr 785 . . . . . . . . 9 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑣 ∈ (metUnif‘𝐷))
169ad3antrrr 721 . . . . . . . . . 10 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑎𝑋)
17 simpllr 793 . . . . . . . . . 10 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑥𝑎)
1816, 17sseldd 3764 . . . . . . . . 9 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑥𝑋)
19 metustbl 22664 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷) ∧ 𝑥𝑋) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏 ⊆ (𝑣 “ {𝑥})))
2014, 15, 18, 19syl3anc 1490 . . . . . . . 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 3771 . . . . . . . . . . 11 ((𝑏 ⊆ (𝑣 “ {𝑥}) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑏𝑎)
2221expcom 402 . . . . . . . . . 10 ((𝑣 “ {𝑥}) ⊆ 𝑎 → (𝑏 ⊆ (𝑣 “ {𝑥}) → 𝑏𝑎))
2322anim2d 605 . . . . . . . . 9 ((𝑣 “ {𝑥}) ⊆ 𝑎 → ((𝑥𝑏𝑏 ⊆ (𝑣 “ {𝑥})) → (𝑥𝑏𝑏𝑎)))
2423reximdv 3162 . . . . . . . 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 489 . . . . . . . 8 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)
2726r19.21bi 3079 . . . . . . 7 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) → ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)
2825, 27r19.29a 3225 . . . . . 6 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎))
2928ralrimiva 3113 . . . . 5 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → ∀𝑥𝑎𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎))
3012, 29jca 507 . . . 4 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → (𝑎 ran (ball‘𝐷) ∧ ∀𝑥𝑎𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎)))
31 fvex 6392 . . . . . 6 (ball‘𝐷) ∈ V
3231rnex 7302 . . . . 5 ran (ball‘𝐷) ∈ V
33 eltg2 21056 . . . . 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 248 . . 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 468 . . . . . . . 8 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → (𝑎 ran (ball‘𝐷) ∧ ∀𝑥𝑎𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎)))
3837simpld 488 . . . . . . 7 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎 ran (ball‘𝐷))
3910ad2antlr 718 . . . . . . 7 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → ran (ball‘𝐷) = 𝑋)
4038, 39sseqtrd 3803 . . . . . 6 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎𝑋)
41 elpwg 4325 . . . . . . 7 (𝑎 ∈ (topGen‘ran (ball‘𝐷)) → (𝑎 ∈ 𝒫 𝑋𝑎𝑋))
4241adantl 473 . . . . . 6 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → (𝑎 ∈ 𝒫 𝑋𝑎𝑋))
4340, 42mpbird 248 . . . . 5 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎 ∈ 𝒫 𝑋)
44 simpllr 793 . . . . . . . . 9 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥𝑎) → 𝐷 ∈ (PsMet‘𝑋))
4540sselda 3763 . . . . . . . . 9 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥𝑎) → 𝑥𝑋)
4637simprd 489 . . . . . . . . . . 11 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → ∀𝑥𝑎𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎))
4746r19.21bi 3079 . . . . . . . . . 10 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥𝑎) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎))
48 blssexps 22524 . . . . . . . . . . 11 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) → (∃𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎) ↔ ∃𝑑 ∈ ℝ+ (𝑥(ball‘𝐷)𝑑) ⊆ 𝑎))
4944, 45, 48syl2anc 579 . . . . . . . . . 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 223 . . . . . . . . 9 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥𝑎) → ∃𝑑 ∈ ℝ+ (𝑥(ball‘𝐷)𝑑) ⊆ 𝑎)
51 blval2 22660 . . . . . . . . . . . . 13 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋𝑑 ∈ ℝ+) → (𝑥(ball‘𝐷)𝑑) = ((𝐷 “ (0[,)𝑑)) “ {𝑥}))
52513expa 1147 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) ∧ 𝑑 ∈ ℝ+) → (𝑥(ball‘𝐷)𝑑) = ((𝐷 “ (0[,)𝑑)) “ {𝑥}))
5352sseq1d 3794 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) ∧ 𝑑 ∈ ℝ+) → ((𝑥(ball‘𝐷)𝑑) ⊆ 𝑎 ↔ ((𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
5453rexbidva 3196 . . . . . . . . . 10 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) → (∃𝑑 ∈ ℝ+ (𝑥(ball‘𝐷)𝑑) ⊆ 𝑎 ↔ ∃𝑑 ∈ ℝ+ ((𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
5554biimpa 468 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) ∧ ∃𝑑 ∈ ℝ+ (𝑥(ball‘𝐷)𝑑) ⊆ 𝑎) → ∃𝑑 ∈ ℝ+ ((𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎)
5644, 45, 50, 55syl21anc 866 . . . . . . . 8 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥𝑎) → ∃𝑑 ∈ ℝ+ ((𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎)
57 cnvexg 7314 . . . . . . . . . . 11 (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ∈ V)
58 imaexg 7305 . . . . . . . . . . 11 (𝐷 ∈ V → (𝐷 “ (0[,)𝑑)) ∈ V)
5957, 58syl 17 . . . . . . . . . 10 (𝐷 ∈ (PsMet‘𝑋) → (𝐷 “ (0[,)𝑑)) ∈ V)
6059ralrimivw 3114 . . . . . . . . 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 5645 . . . . . . . . . . 11 (𝑣 = (𝐷 “ (0[,)𝑑)) → (𝑣 “ {𝑥}) = ((𝐷 “ (0[,)𝑑)) “ {𝑥}))
6362sseq1d 3794 . . . . . . . . . 10 (𝑣 = (𝐷 “ (0[,)𝑑)) → ((𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ((𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
6461, 63rexrnmpt 6563 . . . . . . . . 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 248 . . . . . . 7 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥𝑎) → ∃𝑣 ∈ ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))(𝑣 “ {𝑥}) ⊆ 𝑎)
67 oveq2 6854 . . . . . . . . . . . . . . 15 (𝑑 = 𝑒 → (0[,)𝑑) = (0[,)𝑒))
6867imaeq2d 5650 . . . . . . . . . . . . . 14 (𝑑 = 𝑒 → (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑒)))
6968cbvmptv 4911 . . . . . . . . . . . . 13 (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑))) = (𝑒 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑒)))
7069rneqi 5522 . . . . . . . . . . . 12 ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑))) = ran (𝑒 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑒)))
7170metustfbas 22655 . . . . . . . . . . 11 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑))) ∈ (fBas‘(𝑋 × 𝑋)))
72 ssfg 21969 . . . . . . . . . . 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 22647 . . . . . . . . . . 11 (𝐷 ∈ (PsMet‘𝑋) → (metUnif‘𝐷) = ((𝑋 × 𝑋)filGenran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))))
7574adantl 473 . . . . . . . . . 10 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (metUnif‘𝐷) = ((𝑋 × 𝑋)filGenran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))))
7673, 75sseqtr4d 3804 . . . . . . . . 9 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑))) ⊆ (metUnif‘𝐷))
77 ssrexv 3829 . . . . . . . . 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 717 . . . . . . 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 3113 . . . . 5 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)
8243, 81jca 507 . . . 4 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎))
836biimpar 469 . . . 4 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)) → 𝑎 ∈ (unifTop‘(metUnif‘𝐷)))
8482, 83syldan 585 . . 3 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎 ∈ (unifTop‘(metUnif‘𝐷)))
8535, 84impbida 835 . 2 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑎 ∈ (unifTop‘(metUnif‘𝐷)) ↔ 𝑎 ∈ (topGen‘ran (ball‘𝐷))))
8685eqrdv 2763 1 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (unifTop‘(metUnif‘𝐷)) = (topGen‘ran (ball‘𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  {crab 3059  Vcvv 3350  wss 3734  c0 4081  𝒫 cpw 4317  {csn 4336   cuni 4596  cmpt 4890   × cxp 5277  ccnv 5278  ran crn 5280  cima 5282  cfv 6070  (class class class)co 6846  0cc0 10193  +crp 12033  [,)cico 12384  topGenctg 16378  PsMetcpsmet 20017  ballcbl 20020  fBascfbas 20021  filGencfg 20022  metUnifcmetu 20024  UnifOncust 22296  unifTopcutop 22327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270  ax-pre-sup 10271
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-om 7268  df-1st 7370  df-2nd 7371  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-er 7951  df-map 8066  df-en 8165  df-dom 8166  df-sdom 8167  df-sup 8559  df-inf 8560  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10526  df-neg 10527  df-div 10943  df-nn 11279  df-2 11339  df-n0 11543  df-z 11629  df-uz 11892  df-q 11995  df-rp 12034  df-xneg 12151  df-xadd 12152  df-xmul 12153  df-ico 12388  df-topgen 16384  df-psmet 20025  df-bl 20028  df-fbas 20030  df-fg 20031  df-metu 20032  df-fil 21943  df-ust 22297  df-utop 22328
This theorem is referenced by:  xmetutop  22666
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