Theorem xmetunirn 22052

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 6632	. . . . . 6 ⊢ (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∈ V
2	1	rabex 4773	. . . . 5 ⊢ {𝑑 ∈ (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))} ∈ V
3		df-xmet 19658	. . . . 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 5979	. . . 4 ⊢ ∞Met Fn V
5		fnunirn 6465	. . . 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 22050	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑥) → 𝑥 = dom dom 𝐷)
9	8	fveq2d 6152	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑥) → (∞Met‘𝑥) = (∞Met‘dom dom 𝐷))
10	7, 9	eleqtrd 2700	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑥) → 𝐷 ∈ (∞Met‘dom dom 𝐷))
11	10	rexlimivw 3022	. . 3 ⊢ (∃𝑥 ∈ V 𝐷 ∈ (∞Met‘𝑥) → 𝐷 ∈ (∞Met‘dom dom 𝐷))
12	6, 11	sylbi 207	. 2 ⊢ (𝐷 ∈ ∪ ran ∞Met → 𝐷 ∈ (∞Met‘dom dom 𝐷))
13		fvssunirn 6174	. . 3 ⊢ (∞Met‘dom dom 𝐷) ⊆ ∪ ran ∞Met
14	13	sseli 3579	. 2 ⊢ (𝐷 ∈ (∞Met‘dom dom 𝐷) → 𝐷 ∈ ∪ ran ∞Met)
15	12, 14	impbii 199	1 ⊢ (𝐷 ∈ ∪ ran ∞Met ↔ 𝐷 ∈ (∞Met‘dom dom 𝐷))

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∀wral 2907  ∃wrex 2908  {crab 2911  Vcvv 3186  ∪ cuni 4402   class class class wbr 4613   × cxp 5072  dom cdm 5074  ran crn 5075   Fn wfn 5842  ‘cfv 5847  (class class class)co 6604   ↑_𝑚 cmap 7802  0cc0 9880  ℝ^*cxr 10017   ≤ cle 10019   +_𝑒 cxad 11888  ∞Metcxmt 19650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-xr 10022  df-xmet 19658

This theorem is referenced by:  isxms2  22163  setsmstopn  22193  tngtopn  22364  cfili  22974  cfilfcls  22980