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Theorem metuval 24578
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 variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 df-metu 21381 . 2 metUnif = (𝑑 ran PsMet ↦ ((dom dom 𝑑 × dom dom 𝑑)filGenran (𝑎 ∈ ℝ+ ↦ (𝑑 “ (0[,)𝑎)))))
2 simpr 484 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
32dmeqd 5919 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom 𝑑 = dom 𝐷)
43dmeqd 5919 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = dom dom 𝐷)
5 psmetdmdm 24331 . . . . . 6 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷)
65adantr 480 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑋 = dom dom 𝐷)
74, 6eqtr4d 2778 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = 𝑋)
87sqxpeqd 5721 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (dom dom 𝑑 × dom dom 𝑑) = (𝑋 × 𝑋))
9 simplr 769 . . . . . . 7 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ ℝ+) → 𝑑 = 𝐷)
109cnveqd 5889 . . . . . 6 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ ℝ+) → 𝑑 = 𝐷)
1110imaeq1d 6079 . . . . 5 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ ℝ+) → (𝑑 “ (0[,)𝑎)) = (𝐷 “ (0[,)𝑎)))
1211mpteq2dva 5248 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑎 ∈ ℝ+ ↦ (𝑑 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))))
1312rneqd 5952 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → ran (𝑎 ∈ ℝ+ ↦ (𝑑 “ (0[,)𝑎))) = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))))
148, 13oveq12d 7449 . 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)))))
15 elfvunirn 6939 . 2 (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ran PsMet)
16 ovexd 7466 . 2 (𝐷 ∈ (PsMet‘𝑋) → ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))) ∈ V)
171, 14, 15, 16fvmptd2 7024 1 (𝐷 ∈ (PsMet‘𝑋) → (metUnif‘𝐷) = ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478   cuni 4912  cmpt 5231   × cxp 5687  ccnv 5688  dom cdm 5689  ran crn 5690  cima 5692  cfv 6563  (class class class)co 7431  0cc0 11153  +crp 13032  [,)cico 13386  PsMetcpsmet 21366  filGencfg 21371  metUnifcmetu 21373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-xr 11297  df-psmet 21374  df-metu 21381
This theorem is referenced by:  metuust  24589  cfilucfil2  24590  metuel  24593  psmetutop  24596  restmetu  24599  metucn  24600
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