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Theorem metuval 22762
Description: Value of the uniform structure generated by metric 𝐷. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
Assertion
Ref Expression
metuval (𝐷 ∈ (PsMet‘𝑋) → (metUnif‘𝐷) = ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))))
Distinct variable groups:   𝐷,𝑎   𝑋,𝑎

Proof of Theorem metuval
Dummy variables 𝑢 𝑑 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-metu 20141 . . 3 metUnif = (𝑑 ran PsMet ↦ ((dom dom 𝑑 × dom dom 𝑑)filGenran (𝑎 ∈ ℝ+ ↦ (𝑑 “ (0[,)𝑎)))))
21a1i 11 . 2 (𝐷 ∈ (PsMet‘𝑋) → metUnif = (𝑑 ran PsMet ↦ ((dom dom 𝑑 × dom dom 𝑑)filGenran (𝑎 ∈ ℝ+ ↦ (𝑑 “ (0[,)𝑎))))))
3 simpr 479 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
43dmeqd 5571 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom 𝑑 = dom 𝐷)
54dmeqd 5571 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = dom dom 𝐷)
6 psmetdmdm 22518 . . . . . 6 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷)
76adantr 474 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑋 = dom dom 𝐷)
85, 7eqtr4d 2816 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = 𝑋)
98sqxpeqd 5387 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (dom dom 𝑑 × dom dom 𝑑) = (𝑋 × 𝑋))
10 simplr 759 . . . . . . 7 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ ℝ+) → 𝑑 = 𝐷)
1110cnveqd 5543 . . . . . 6 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ ℝ+) → 𝑑 = 𝐷)
1211imaeq1d 5719 . . . . 5 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ ℝ+) → (𝑑 “ (0[,)𝑎)) = (𝐷 “ (0[,)𝑎)))
1312mpteq2dva 4979 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑎 ∈ ℝ+ ↦ (𝑑 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))))
1413rneqd 5598 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → ran (𝑎 ∈ ℝ+ ↦ (𝑑 “ (0[,)𝑎))) = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))))
159, 14oveq12d 6940 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → ((dom dom 𝑑 × dom dom 𝑑)filGenran (𝑎 ∈ ℝ+ ↦ (𝑑 “ (0[,)𝑎)))) = ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))))
16 elfvdm 6478 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ dom PsMet)
17 fveq2 6446 . . . . . 6 (𝑥 = 𝑋 → (PsMet‘𝑥) = (PsMet‘𝑋))
1817eleq2d 2844 . . . . 5 (𝑥 = 𝑋 → (𝐷 ∈ (PsMet‘𝑥) ↔ 𝐷 ∈ (PsMet‘𝑋)))
1918rspcev 3510 . . . 4 ((𝑋 ∈ dom PsMet ∧ 𝐷 ∈ (PsMet‘𝑋)) → ∃𝑥 ∈ dom PsMet𝐷 ∈ (PsMet‘𝑥))
2016, 19mpancom 678 . . 3 (𝐷 ∈ (PsMet‘𝑋) → ∃𝑥 ∈ dom PsMet𝐷 ∈ (PsMet‘𝑥))
21 df-psmet 20134 . . . . 5 PsMet = (𝑦 ∈ V ↦ {𝑢 ∈ (ℝ*𝑚 (𝑦 × 𝑦)) ∣ ∀𝑧𝑦 ((𝑧𝑢𝑧) = 0 ∧ ∀𝑤𝑦𝑣𝑦 (𝑧𝑢𝑤) ≤ ((𝑣𝑢𝑧) +𝑒 (𝑣𝑢𝑤)))})
2221funmpt2 6174 . . . 4 Fun PsMet
23 elunirn 6781 . . . 4 (Fun PsMet → (𝐷 ran PsMet ↔ ∃𝑥 ∈ dom PsMet𝐷 ∈ (PsMet‘𝑥)))
2422, 23ax-mp 5 . . 3 (𝐷 ran PsMet ↔ ∃𝑥 ∈ dom PsMet𝐷 ∈ (PsMet‘𝑥))
2520, 24sylibr 226 . 2 (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ran PsMet)
26 ovexd 6956 . 2 (𝐷 ∈ (PsMet‘𝑋) → ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))) ∈ V)
272, 15, 25, 26fvmptd 6548 1 (𝐷 ∈ (PsMet‘𝑋) → (metUnif‘𝐷) = ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wcel 2106  wral 3089  wrex 3090  {crab 3093  Vcvv 3397   cuni 4671   class class class wbr 4886  cmpt 4965   × cxp 5353  ccnv 5354  dom cdm 5355  ran crn 5356  cima 5358  Fun wfun 6129  cfv 6135  (class class class)co 6922  𝑚 cmap 8140  0cc0 10272  *cxr 10410  cle 10412  +crp 12137   +𝑒 cxad 12255  [,)cico 12489  PsMetcpsmet 20126  filGencfg 20131  metUnifcmetu 20133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-map 8142  df-xr 10415  df-psmet 20134  df-metu 20141
This theorem is referenced by:  metuust  22773  cfilucfil2  22774  metuel  22777  psmetutop  22780  restmetu  22783  metucn  22784
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