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Theorem psmetutop 24595
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 24588 . . . . . . . . . . . 12 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (metUnif‘𝐷) ∈ (UnifOn‘𝑋))
2 utopval 24256 . . . . . . . . . . . 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 2824 . . . . . . . . . 10 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑎 ∈ (unifTop‘(metUnif‘𝐷)) ↔ 𝑎 ∈ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎}))
5 rabid 3454 . . . . . . . . . 10 (𝑎 ∈ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎} ↔ (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎))
64, 5bitrdi 287 . . . . . . . . 9 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑎 ∈ (unifTop‘(metUnif‘𝐷)) ↔ (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)))
76biimpa 476 . . . . . . . 8 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎))
87simpld 494 . . . . . . 7 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → 𝑎 ∈ 𝒫 𝑋)
98elpwid 4613 . . . . . 6 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → 𝑎𝑋)
10 unirnblps 24444 . . . . . . 7 (𝐷 ∈ (PsMet‘𝑋) → ran (ball‘𝐷) = 𝑋)
1110ad2antlr 727 . . . . . 6 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → ran (ball‘𝐷) = 𝑋)
129, 11sseqtrrd 4036 . . . . 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))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → (𝑣 “ {𝑥}) ⊆ 𝑎)
14 simp-5r 786 . . . . . . . . 9 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝐷 ∈ (PsMet‘𝑋))
15 simplr 769 . . . . . . . . 9 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑣 ∈ (metUnif‘𝐷))
169ad3antrrr 730 . . . . . . . . . 10 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑎𝑋)
17 simpllr 776 . . . . . . . . . 10 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑥𝑎)
1816, 17sseldd 3995 . . . . . . . . 9 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑥𝑋)
19 metustbl 24594 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷) ∧ 𝑥𝑋) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏 ⊆ (𝑣 “ {𝑥})))
2014, 15, 18, 19syl3anc 1370 . . . . . . . 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 4003 . . . . . . . . . . 11 ((𝑏 ⊆ (𝑣 “ {𝑥}) ∧ (𝑣 “ {𝑥}) ⊆ 𝑎) → 𝑏𝑎)
2221expcom 413 . . . . . . . . . 10 ((𝑣 “ {𝑥}) ⊆ 𝑎 → (𝑏 ⊆ (𝑣 “ {𝑥}) → 𝑏𝑎))
2322anim2d 612 . . . . . . . . 9 ((𝑣 “ {𝑥}) ⊆ 𝑎 → ((𝑥𝑏𝑏 ⊆ (𝑣 “ {𝑥})) → (𝑥𝑏𝑏𝑎)))
2423reximdv 3167 . . . . . . . 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 495 . . . . . . . 8 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)
2726r19.21bi 3248 . . . . . . 7 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) → ∃𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)
2825, 27r19.29a 3159 . . . . . 6 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) ∧ 𝑥𝑎) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎))
2928ralrimiva 3143 . . . . 5 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → ∀𝑥𝑎𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎))
3012, 29jca 511 . . . 4 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (unifTop‘(metUnif‘𝐷))) → (𝑎 ran (ball‘𝐷) ∧ ∀𝑥𝑎𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎)))
31 fvex 6919 . . . . . 6 (ball‘𝐷) ∈ V
3231rnex 7932 . . . . 5 ran (ball‘𝐷) ∈ V
33 eltg2 22980 . . . . 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 257 . . 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 476 . . . . . . . 8 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → (𝑎 ran (ball‘𝐷) ∧ ∀𝑥𝑎𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎)))
3837simpld 494 . . . . . . 7 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎 ran (ball‘𝐷))
3910ad2antlr 727 . . . . . . 7 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → ran (ball‘𝐷) = 𝑋)
4038, 39sseqtrd 4035 . . . . . 6 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → 𝑎𝑋)
41 elpwg 4607 . . . . . . 7 (𝑎 ∈ (topGen‘ran (ball‘𝐷)) → (𝑎 ∈ 𝒫 𝑋𝑎𝑋))
4241adantl 481 . . . . . 6 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → (𝑎 ∈ 𝒫 𝑋𝑎𝑋))
4340, 42mpbird 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))) ∧ 𝑥𝑎) → 𝐷 ∈ (PsMet‘𝑋))
4540sselda 3994 . . . . . . . . 9 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥𝑎) → 𝑥𝑋)
4637simprd 495 . . . . . . . . . . 11 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → ∀𝑥𝑎𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎))
4746r19.21bi 3248 . . . . . . . . . 10 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥𝑎) → ∃𝑏 ∈ ran (ball‘𝐷)(𝑥𝑏𝑏𝑎))
48 blssexps 24451 . . . . . . . . . . 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 232 . . . . . . . . 9 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥𝑎) → ∃𝑑 ∈ ℝ+ (𝑥(ball‘𝐷)𝑑) ⊆ 𝑎)
51 blval2 24590 . . . . . . . . . . . . 13 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋𝑑 ∈ ℝ+) → (𝑥(ball‘𝐷)𝑑) = ((𝐷 “ (0[,)𝑑)) “ {𝑥}))
52513expa 1117 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) ∧ 𝑑 ∈ ℝ+) → (𝑥(ball‘𝐷)𝑑) = ((𝐷 “ (0[,)𝑑)) “ {𝑥}))
5352sseq1d 4026 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) ∧ 𝑑 ∈ ℝ+) → ((𝑥(ball‘𝐷)𝑑) ⊆ 𝑎 ↔ ((𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
5453rexbidva 3174 . . . . . . . . . 10 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) → (∃𝑑 ∈ ℝ+ (𝑥(ball‘𝐷)𝑑) ⊆ 𝑎 ↔ ∃𝑑 ∈ ℝ+ ((𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
5554biimpa 476 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋) ∧ ∃𝑑 ∈ ℝ+ (𝑥(ball‘𝐷)𝑑) ⊆ 𝑎) → ∃𝑑 ∈ ℝ+ ((𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎)
5644, 45, 50, 55syl21anc 838 . . . . . . . 8 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥𝑎) → ∃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)
5957, 58syl 17 . . . . . . . . . 10 (𝐷 ∈ (PsMet‘𝑋) → (𝐷 “ (0[,)𝑑)) ∈ V)
6059ralrimivw 3147 . . . . . . . . 9 (𝐷 ∈ (PsMet‘𝑋) → ∀𝑑 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) ∈ V)
61 eqid 2734 . . . . . . . . . 10 (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑))) = (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))
62 imaeq1 6074 . . . . . . . . . . 11 (𝑣 = (𝐷 “ (0[,)𝑑)) → (𝑣 “ {𝑥}) = ((𝐷 “ (0[,)𝑑)) “ {𝑥}))
6362sseq1d 4026 . . . . . . . . . 10 (𝑣 = (𝐷 “ (0[,)𝑑)) → ((𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ((𝐷 “ (0[,)𝑑)) “ {𝑥}) ⊆ 𝑎))
6461, 63rexrnmptw 7114 . . . . . . . . 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 257 . . . . . . 7 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) ∧ 𝑥𝑎) → ∃𝑣 ∈ ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))(𝑣 “ {𝑥}) ⊆ 𝑎)
67 oveq2 7438 . . . . . . . . . . . . . . 15 (𝑑 = 𝑒 → (0[,)𝑑) = (0[,)𝑒))
6867imaeq2d 6079 . . . . . . . . . . . . . 14 (𝑑 = 𝑒 → (𝐷 “ (0[,)𝑑)) = (𝐷 “ (0[,)𝑒)))
6968cbvmptv 5260 . . . . . . . . . . . . 13 (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑))) = (𝑒 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑒)))
7069rneqi 5950 . . . . . . . . . . . 12 ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑))) = ran (𝑒 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑒)))
7170metustfbas 24585 . . . . . . . . . . 11 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑))) ∈ (fBas‘(𝑋 × 𝑋)))
72 ssfg 23895 . . . . . . . . . . 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 24577 . . . . . . . . . . 11 (𝐷 ∈ (PsMet‘𝑋) → (metUnif‘𝐷) = ((𝑋 × 𝑋)filGenran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))))
7574adantl 481 . . . . . . . . . 10 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (metUnif‘𝐷) = ((𝑋 × 𝑋)filGenran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))))
7673, 75sseqtrrd 4036 . . . . . . . . 9 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑))) ⊆ (metUnif‘𝐷))
77 ssrexv 4064 . . . . . . . . 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 726 . . . . . . 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 3143 . . . . 5 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎)
8243, 81jca 511 . . . 4 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑎 ∈ (topGen‘ran (ball‘𝐷))) → (𝑎 ∈ 𝒫 𝑋 ∧ ∀𝑥𝑎𝑣 ∈ (metUnif‘𝐷)(𝑣 “ {𝑥}) ⊆ 𝑎))
836biimpar 477 . . . 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 801 . 2 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑎 ∈ (unifTop‘(metUnif‘𝐷)) ↔ 𝑎 ∈ (topGen‘ran (ball‘𝐷))))
8685eqrdv 2732 1 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (unifTop‘(metUnif‘𝐷)) = (topGen‘ran (ball‘𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wne 2937  wral 3058  wrex 3067  {crab 3432  Vcvv 3477  wss 3962  c0 4338  𝒫 cpw 4604  {csn 4630   cuni 4911  cmpt 5230   × cxp 5686  ccnv 5687  ran crn 5689  cima 5691  cfv 6562  (class class class)co 7430  0cc0 11152  +crp 13031  [,)cico 13385  topGenctg 17483  PsMetcpsmet 21365  ballcbl 21368  fBascfbas 21369  filGencfg 21370  metUnifcmetu 21372  UnifOncust 24223  unifTopcutop 24254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-sup 9479  df-inf 9480  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-n0 12524  df-z 12611  df-uz 12876  df-q 12988  df-rp 13032  df-xneg 13151  df-xadd 13152  df-xmul 13153  df-ico 13389  df-topgen 17489  df-psmet 21373  df-bl 21376  df-fbas 21378  df-fg 21379  df-metu 21380  df-fil 23869  df-ust 24224  df-utop 24255
This theorem is referenced by:  xmetutop  24596
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