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Theorem xmetunirn 21952
 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 6577 . . . . . 6 (ℝ*𝑚 (𝑥 × 𝑥)) ∈ V
21rabex 4740 . . . . 5 {𝑑 ∈ (ℝ*𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥𝑧𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))} ∈ V
3 df-xmet 19560 . . . . 5 ∞Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥𝑧𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
42, 3fnmpti 5935 . . . 4 ∞Met Fn V
5 fnunirn 6415 . . . 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 21950 . . . . . 6 (𝐷 ∈ (∞Met‘𝑥) → 𝑥 = dom dom 𝐷)
98fveq2d 6107 . . . . 5 (𝐷 ∈ (∞Met‘𝑥) → (∞Met‘𝑥) = (∞Met‘dom dom 𝐷))
107, 9eleqtrd 2690 . . . 4 (𝐷 ∈ (∞Met‘𝑥) → 𝐷 ∈ (∞Met‘dom dom 𝐷))
1110rexlimivw 3011 . . 3 (∃𝑥 ∈ V 𝐷 ∈ (∞Met‘𝑥) → 𝐷 ∈ (∞Met‘dom dom 𝐷))
126, 11sylbi 206 . 2 (𝐷 ran ∞Met → 𝐷 ∈ (∞Met‘dom dom 𝐷))
13 fvssunirn 6127 . . 3 (∞Met‘dom dom 𝐷) ⊆ ran ∞Met
1413sseli 3564 . 2 (𝐷 ∈ (∞Met‘dom dom 𝐷) → 𝐷 ran ∞Met)
1512, 14impbii 198 1 (𝐷 ran ∞Met ↔ 𝐷 ∈ (∞Met‘dom dom 𝐷))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173  ∪ cuni 4372   class class class wbr 4583   × cxp 5036  dom cdm 5038  ran crn 5039   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  0cc0 9815  ℝ*cxr 9952   ≤ cle 9954   +𝑒 cxad 11820  ∞Metcxmt 19552 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-xr 9957  df-xmet 19560 This theorem is referenced by:  isxms2  22063  setsmstopn  22093  tngtopn  22264  cfili  22874  cfilfcls  22880
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