Theorem xmetunirn 15072

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 𝑑 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnmap 6819	. . . . . . 7 ⊢ ↑𝑚 Fn (V × V)
2		xrex 10081	. . . . . . 7 ⊢ ℝ* ∈ V
3		sqxpexg 4841	. . . . . . . 8 ⊢ (𝑥 ∈ V → (𝑥 × 𝑥) ∈ V)
4	3	elv 2804	. . . . . . 7 ⊢ (𝑥 × 𝑥) ∈ V
5		fnovex 6046	. . . . . . 7 ⊢ (( ↑𝑚 Fn (V × V) ∧ ℝ* ∈ V ∧ (𝑥 × 𝑥) ∈ V) → (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∈ V)
6	1, 2, 4, 5	mp3an 1371	. . . . . 6 ⊢ (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∈ V
7	6	rabex 4232	. . . . 5 ⊢ {𝑑 ∈ (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))} ∈ V
8		df-xmet 14548	. . . . 5 ⊢ ∞Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
9	7, 8	fnmpti 5458	. . . 4 ⊢ ∞Met Fn V
10		fnunirn 5903	. . . 4 ⊢ (∞Met Fn V → (𝐷 ∈ ∪ ran ∞Met ↔ ∃𝑥 ∈ V 𝐷 ∈ (∞Met‘𝑥)))
11	9, 10	ax-mp 5	. . 3 ⊢ (𝐷 ∈ ∪ ran ∞Met ↔ ∃𝑥 ∈ V 𝐷 ∈ (∞Met‘𝑥))
12		id 19	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑥) → 𝐷 ∈ (∞Met‘𝑥))
13		xmetdmdm 15070	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑥) → 𝑥 = dom dom 𝐷)
14	13	fveq2d 5639	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑥) → (∞Met‘𝑥) = (∞Met‘dom dom 𝐷))
15	12, 14	eleqtrd 2308	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑥) → 𝐷 ∈ (∞Met‘dom dom 𝐷))
16	15	rexlimivw 2644	. . 3 ⊢ (∃𝑥 ∈ V 𝐷 ∈ (∞Met‘𝑥) → 𝐷 ∈ (∞Met‘dom dom 𝐷))
17	11, 16	sylbi 121	. 2 ⊢ (𝐷 ∈ ∪ ran ∞Met → 𝐷 ∈ (∞Met‘dom dom 𝐷))
18		elex 2812	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘dom dom 𝐷) → 𝐷 ∈ V)
19		dmexg 4994	. . . . . 6 ⊢ (𝐷 ∈ V → dom 𝐷 ∈ V)
20		dmexg 4994	. . . . . 6 ⊢ (dom 𝐷 ∈ V → dom dom 𝐷 ∈ V)
21	18, 19, 20	3syl 17	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘dom dom 𝐷) → dom dom 𝐷 ∈ V)
22		fvssunirng 5650	. . . . 5 ⊢ (dom dom 𝐷 ∈ V → (∞Met‘dom dom 𝐷) ⊆ ∪ ran ∞Met)
23	21, 22	syl 14	. . . 4 ⊢ (𝐷 ∈ (∞Met‘dom dom 𝐷) → (∞Met‘dom dom 𝐷) ⊆ ∪ ran ∞Met)
24	23	sseld 3224	. . 3 ⊢ (𝐷 ∈ (∞Met‘dom dom 𝐷) → (𝐷 ∈ (∞Met‘dom dom 𝐷) → 𝐷 ∈ ∪ ran ∞Met))
25	24	pm2.43i 49	. 2 ⊢ (𝐷 ∈ (∞Met‘dom dom 𝐷) → 𝐷 ∈ ∪ ran ∞Met)
26	17, 25	impbii 126	1 ⊢ (𝐷 ∈ ∪ ran ∞Met ↔ 𝐷 ∈ (∞Met‘dom dom 𝐷))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  {crab 2512  Vcvv 2800   ⊆ wss 3198  ∪ cuni 3891   class class class wbr 4086   × cxp 4721  dom cdm 4723  ran crn 4724   Fn wfn 5319  ‘cfv 5324  (class class class)co 6013   ↑_𝑚 cmap 6812  0cc0 8022  ℝ^*cxr 8203   ≤ cle 8205   +_𝑒 cxad 9995  ∞Metcxmet 14540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-map 6814  df-pnf 8206  df-mnf 8207  df-xr 8208  df-xmet 14548

This theorem is referenced by:  isxms2  15166