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Theorem xmetunirn 22954
 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 7169 . . . . . 6 (ℝ*m (𝑥 × 𝑥)) ∈ V
21rabex 5200 . . . . 5 {𝑑 ∈ (ℝ*m (𝑥 × 𝑥)) ∣ ∀𝑦𝑥𝑧𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))} ∈ V
3 df-xmet 20088 . . . . 5 ∞Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*m (𝑥 × 𝑥)) ∣ ∀𝑦𝑥𝑧𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
42, 3fnmpti 6464 . . . 4 ∞Met Fn V
5 fnunirn 6991 . . . 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 22952 . . . . . 6 (𝐷 ∈ (∞Met‘𝑥) → 𝑥 = dom dom 𝐷)
98fveq2d 6650 . . . . 5 (𝐷 ∈ (∞Met‘𝑥) → (∞Met‘𝑥) = (∞Met‘dom dom 𝐷))
107, 9eleqtrd 2892 . . . 4 (𝐷 ∈ (∞Met‘𝑥) → 𝐷 ∈ (∞Met‘dom dom 𝐷))
1110rexlimivw 3241 . . 3 (∃𝑥 ∈ V 𝐷 ∈ (∞Met‘𝑥) → 𝐷 ∈ (∞Met‘dom dom 𝐷))
126, 11sylbi 220 . 2 (𝐷 ran ∞Met → 𝐷 ∈ (∞Met‘dom dom 𝐷))
13 fvssunirn 6675 . . 3 (∞Met‘dom dom 𝐷) ⊆ ran ∞Met
1413sseli 3911 . 2 (𝐷 ∈ (∞Met‘dom dom 𝐷) → 𝐷 ran ∞Met)
1512, 14impbii 212 1 (𝐷 ran ∞Met ↔ 𝐷 ∈ (∞Met‘dom dom 𝐷))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  {crab 3110  Vcvv 3441  ∪ cuni 4801   class class class wbr 5031   × cxp 5518  dom cdm 5520  ran crn 5521   Fn wfn 6320  ‘cfv 6325  (class class class)co 7136   ↑m cmap 8392  0cc0 10529  ℝ*cxr 10666   ≤ cle 10668   +𝑒 cxad 12496  ∞Metcxmet 20080 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-cnex 10585  ax-resscn 10586 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-map 8394  df-xr 10671  df-xmet 20088 This theorem is referenced by:  isxms2  23065  setsmstopn  23095  tngtopn  23266  cfili  23882  cfilfcls  23888
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