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Theorem xmetunirn 22421
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 6874 . . . . . 6 (ℝ*𝑚 (𝑥 × 𝑥)) ∈ V
21rabex 4973 . . . . 5 {𝑑 ∈ (ℝ*𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥𝑧𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))} ∈ V
3 df-xmet 20012 . . . . 5 ∞Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥𝑧𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
42, 3fnmpti 6200 . . . 4 ∞Met Fn V
5 fnunirn 6703 . . . 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 22419 . . . . . 6 (𝐷 ∈ (∞Met‘𝑥) → 𝑥 = dom dom 𝐷)
98fveq2d 6379 . . . . 5 (𝐷 ∈ (∞Met‘𝑥) → (∞Met‘𝑥) = (∞Met‘dom dom 𝐷))
107, 9eleqtrd 2846 . . . 4 (𝐷 ∈ (∞Met‘𝑥) → 𝐷 ∈ (∞Met‘dom dom 𝐷))
1110rexlimivw 3176 . . 3 (∃𝑥 ∈ V 𝐷 ∈ (∞Met‘𝑥) → 𝐷 ∈ (∞Met‘dom dom 𝐷))
126, 11sylbi 208 . 2 (𝐷 ran ∞Met → 𝐷 ∈ (∞Met‘dom dom 𝐷))
13 fvssunirn 6404 . . 3 (∞Met‘dom dom 𝐷) ⊆ ran ∞Met
1413sseli 3757 . 2 (𝐷 ∈ (∞Met‘dom dom 𝐷) → 𝐷 ran ∞Met)
1512, 14impbii 200 1 (𝐷 ran ∞Met ↔ 𝐷 ∈ (∞Met‘dom dom 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  wrex 3056  {crab 3059  Vcvv 3350   cuni 4594   class class class wbr 4809   × cxp 5275  dom cdm 5277  ran crn 5278   Fn wfn 6063  cfv 6068  (class class class)co 6842  𝑚 cmap 8060  0cc0 10189  *cxr 10327  cle 10329   +𝑒 cxad 12144  ∞Metcxmet 20004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-map 8062  df-xr 10332  df-xmet 20012
This theorem is referenced by:  isxms2  22532  setsmstopn  22562  tngtopn  22733  cfili  23345  cfilfcls  23351
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