Theorem xmetunirn 15169

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 6867	. . . . . . 7 ⊢ ↑𝑚 Fn (V × V)
2		xrex 10152	. . . . . . 7 ⊢ ℝ* ∈ V
3		sqxpexg 4849	. . . . . . . 8 ⊢ (𝑥 ∈ V → (𝑥 × 𝑥) ∈ V)
4	3	elv 2807	. . . . . . 7 ⊢ (𝑥 × 𝑥) ∈ V
5		fnovex 6061	. . . . . . 7 ⊢ (( ↑𝑚 Fn (V × V) ∧ ℝ* ∈ V ∧ (𝑥 × 𝑥) ∈ V) → (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∈ V)
6	1, 2, 4, 5	mp3an 1374	. . . . . 6 ⊢ (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∈ V
7	6	rabex 4239	. . . . 5 ⊢ {𝑑 ∈ (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))} ∈ V
8		df-xmet 14640	. . . . 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 5468	. . . 4 ⊢ ∞Met Fn V
10		fnunirn 5918	. . . 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 15167	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑥) → 𝑥 = dom dom 𝐷)
14	13	fveq2d 5652	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑥) → (∞Met‘𝑥) = (∞Met‘dom dom 𝐷))
15	12, 14	eleqtrd 2310	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑥) → 𝐷 ∈ (∞Met‘dom dom 𝐷))
16	15	rexlimivw 2647	. . 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 2815	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘dom dom 𝐷) → 𝐷 ∈ V)
19		dmexg 5002	. . . . . 6 ⊢ (𝐷 ∈ V → dom 𝐷 ∈ V)
20		dmexg 5002	. . . . . 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 5663	. . . . 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 3227	. . 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 1398   ∈ wcel 2202  ∀wral 2511  ∃wrex 2512  {crab 2515  Vcvv 2803   ⊆ wss 3201  ∪ cuni 3898   class class class wbr 4093   × cxp 4729  dom cdm 4731  ran crn 4732   Fn wfn 5328  ‘cfv 5333  (class class class)co 6028   ↑_𝑚 cmap 6860  0cc0 8092  ℝ^*cxr 8272   ≤ cle 8274   +_𝑒 cxad 10066  ∞Metcxmet 14632

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-map 6862  df-pnf 8275  df-mnf 8276  df-xr 8277  df-xmet 14640

This theorem is referenced by:  isxms2  15263