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Theorem xmetunirn 22874
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 7178 . . . . . 6 (ℝ*m (𝑥 × 𝑥)) ∈ V
21rabex 5226 . . . . 5 {𝑑 ∈ (ℝ*m (𝑥 × 𝑥)) ∣ ∀𝑦𝑥𝑧𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))} ∈ V
3 df-xmet 20466 . . . . 5 ∞Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*m (𝑥 × 𝑥)) ∣ ∀𝑦𝑥𝑧𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
42, 3fnmpti 6484 . . . 4 ∞Met Fn V
5 fnunirn 7003 . . . 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 22872 . . . . . 6 (𝐷 ∈ (∞Met‘𝑥) → 𝑥 = dom dom 𝐷)
98fveq2d 6667 . . . . 5 (𝐷 ∈ (∞Met‘𝑥) → (∞Met‘𝑥) = (∞Met‘dom dom 𝐷))
107, 9eleqtrd 2912 . . . 4 (𝐷 ∈ (∞Met‘𝑥) → 𝐷 ∈ (∞Met‘dom dom 𝐷))
1110rexlimivw 3279 . . 3 (∃𝑥 ∈ V 𝐷 ∈ (∞Met‘𝑥) → 𝐷 ∈ (∞Met‘dom dom 𝐷))
126, 11sylbi 218 . 2 (𝐷 ran ∞Met → 𝐷 ∈ (∞Met‘dom dom 𝐷))
13 fvssunirn 6692 . . 3 (∞Met‘dom dom 𝐷) ⊆ ran ∞Met
1413sseli 3960 . 2 (𝐷 ∈ (∞Met‘dom dom 𝐷) → 𝐷 ran ∞Met)
1512, 14impbii 210 1 (𝐷 ran ∞Met ↔ 𝐷 ∈ (∞Met‘dom dom 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  {crab 3139  Vcvv 3492   cuni 4830   class class class wbr 5057   × cxp 5546  dom cdm 5548  ran crn 5549   Fn wfn 6343  cfv 6348  (class class class)co 7145  m cmap 8395  0cc0 10525  *cxr 10662  cle 10664   +𝑒 cxad 12493  ∞Metcxmet 20458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-xr 10667  df-xmet 20466
This theorem is referenced by:  isxms2  22985  setsmstopn  23015  tngtopn  23186  cfili  23798  cfilfcls  23804
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