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Theorem xmetunirn 23642
Description: Two ways to express an extended metric on an unspecified base. (Contributed by Mario Carneiro, 13-Oct-2015.)
Assertion
Ref Expression
xmetunirn (𝐷 ran ∞Met ↔ 𝐷 ∈ (∞Met‘dom dom 𝐷))

Proof of Theorem xmetunirn
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 7384 . . . . . 6 (ℝ*m (𝑥 × 𝑥)) ∈ V
21rabex 5287 . . . . 5 {𝑑 ∈ (ℝ*m (𝑥 × 𝑥)) ∣ ∀𝑦𝑥𝑧𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))} ∈ V
3 df-xmet 20742 . . . . 5 ∞Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*m (𝑥 × 𝑥)) ∣ ∀𝑦𝑥𝑧𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
42, 3fnmpti 6641 . . . 4 ∞Met Fn V
5 fnunirn 7197 . . . 4 (∞Met Fn V → (𝐷 ran ∞Met ↔ ∃𝑥 ∈ V 𝐷 ∈ (∞Met‘𝑥)))
64, 5ax-mp 5 . . 3 (𝐷 ran ∞Met ↔ ∃𝑥 ∈ V 𝐷 ∈ (∞Met‘𝑥))
7 id 22 . . . . 5 (𝐷 ∈ (∞Met‘𝑥) → 𝐷 ∈ (∞Met‘𝑥))
8 xmetdmdm 23640 . . . . . 6 (𝐷 ∈ (∞Met‘𝑥) → 𝑥 = dom dom 𝐷)
98fveq2d 6843 . . . . 5 (𝐷 ∈ (∞Met‘𝑥) → (∞Met‘𝑥) = (∞Met‘dom dom 𝐷))
107, 9eleqtrd 2840 . . . 4 (𝐷 ∈ (∞Met‘𝑥) → 𝐷 ∈ (∞Met‘dom dom 𝐷))
1110rexlimivw 3146 . . 3 (∃𝑥 ∈ V 𝐷 ∈ (∞Met‘𝑥) → 𝐷 ∈ (∞Met‘dom dom 𝐷))
126, 11sylbi 216 . 2 (𝐷 ran ∞Met → 𝐷 ∈ (∞Met‘dom dom 𝐷))
13 fvssunirn 6872 . . 3 (∞Met‘dom dom 𝐷) ⊆ ran ∞Met
1413sseli 3938 . 2 (𝐷 ∈ (∞Met‘dom dom 𝐷) → 𝐷 ran ∞Met)
1512, 14impbii 208 1 (𝐷 ran ∞Met ↔ 𝐷 ∈ (∞Met‘dom dom 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3062  wrex 3071  {crab 3405  Vcvv 3443   cuni 4863   class class class wbr 5103   × cxp 5629  dom cdm 5631  ran crn 5632   Fn wfn 6488  cfv 6493  (class class class)co 7351  m cmap 8723  0cc0 11009  *cxr 11146  cle 11148   +𝑒 cxad 12985  ∞Metcxmet 20734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664  ax-cnex 11065  ax-resscn 11066
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-map 8725  df-xr 11151  df-xmet 20742
This theorem is referenced by:  isxms2  23753  setsmstopn  23785  tngtopn  23966  cfili  24584  cfilfcls  24590
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