Theorem xmetunirn 24369

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.

Step	Hyp	Ref	Expression
1		ovex 7468	. . . . . 6 ⊢ (ℝ* ↑m (𝑥 × 𝑥)) ∈ V
2	1	rabex 5346	. . . . 5 ⊢ {𝑑 ∈ (ℝ* ↑m (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))} ∈ V
3		df-xmet 21381	. . . . 5 ⊢ ∞Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ* ↑m (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
4	2, 3	fnmpti 6716	. . . 4 ⊢ ∞Met Fn V
5		fnunirn 7278	. . . 4 ⊢ (∞Met Fn V → (𝐷 ∈ ∪ ran ∞Met ↔ ∃𝑥 ∈ V 𝐷 ∈ (∞Met‘𝑥)))
6	4, 5	ax-mp 5	. . 3 ⊢ (𝐷 ∈ ∪ ran ∞Met ↔ ∃𝑥 ∈ V 𝐷 ∈ (∞Met‘𝑥))
7		id 22	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑥) → 𝐷 ∈ (∞Met‘𝑥))
8		xmetdmdm 24367	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑥) → 𝑥 = dom dom 𝐷)
9	8	fveq2d 6915	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑥) → (∞Met‘𝑥) = (∞Met‘dom dom 𝐷))
10	7, 9	eleqtrd 2842	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑥) → 𝐷 ∈ (∞Met‘dom dom 𝐷))
11	10	rexlimivw 3150	. . 3 ⊢ (∃𝑥 ∈ V 𝐷 ∈ (∞Met‘𝑥) → 𝐷 ∈ (∞Met‘dom dom 𝐷))
12	6, 11	sylbi 217	. 2 ⊢ (𝐷 ∈ ∪ ran ∞Met → 𝐷 ∈ (∞Met‘dom dom 𝐷))
13		fvssunirn 6944	. . 3 ⊢ (∞Met‘dom dom 𝐷) ⊆ ∪ ran ∞Met
14	13	sseli 3992	. 2 ⊢ (𝐷 ∈ (∞Met‘dom dom 𝐷) → 𝐷 ∈ ∪ ran ∞Met)
15	12, 14	impbii 209	1 ⊢ (𝐷 ∈ ∪ ran ∞Met ↔ 𝐷 ∈ (∞Met‘dom dom 𝐷))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1538   ∈ wcel 2107  ∀wral 3060  ∃wrex 3069  {crab 3434  Vcvv 3479  ∪ cuni 4913   class class class wbr 5149   × cxp 5688  dom cdm 5690  ran crn 5691   Fn wfn 6561  ‘cfv 6566  (class class class)co 7435   ↑_m cmap 8871  0cc0 11159  ℝ^*cxr 11298   ≤ cle 11300   +_𝑒 cxad 13156  ∞Metcxmet 21373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5303  ax-nul 5313  ax-pow 5372  ax-pr 5439  ax-un 7758  ax-cnex 11215  ax-resscn 11216

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1541  df-fal 1551  df-ex 1778  df-nf 1782  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3435  df-v 3481  df-sbc 3793  df-dif 3967  df-un 3969  df-in 3971  df-ss 3981  df-nul 4341  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4914  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5584  df-xp 5696  df-rel 5697  df-cnv 5698  df-co 5699  df-dm 5700  df-rn 5701  df-iota 6519  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-ov 7438  df-oprab 7439  df-mpo 7440  df-map 8873  df-xr 11303  df-xmet 21381

This theorem is referenced by:  isxms2  24480  setsmstopn  24512  tngtopn  24693  cfili  25324  cfilfcls  25330