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Mirrors > Home > ILE Home > Th. List > xmetunirn | GIF version |
Description: Two ways to express an extended metric on an unspecified base. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Ref | Expression |
---|---|
xmetunirn | ⊢ (𝐷 ∈ ∪ ran ∞Met ↔ 𝐷 ∈ (∞Met‘dom dom 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnmap 6549 | . . . . . . 7 ⊢ ↑𝑚 Fn (V × V) | |
2 | xrex 9639 | . . . . . . 7 ⊢ ℝ* ∈ V | |
3 | sqxpexg 4655 | . . . . . . . 8 ⊢ (𝑥 ∈ V → (𝑥 × 𝑥) ∈ V) | |
4 | 3 | elv 2690 | . . . . . . 7 ⊢ (𝑥 × 𝑥) ∈ V |
5 | fnovex 5804 | . . . . . . 7 ⊢ (( ↑𝑚 Fn (V × V) ∧ ℝ* ∈ V ∧ (𝑥 × 𝑥) ∈ V) → (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∈ V) | |
6 | 1, 2, 4, 5 | mp3an 1315 | . . . . . 6 ⊢ (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∈ V |
7 | 6 | rabex 4072 | . . . . 5 ⊢ {𝑑 ∈ (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))} ∈ V |
8 | df-xmet 12157 | . . . . 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 5251 | . . . 4 ⊢ ∞Met Fn V |
10 | fnunirn 5668 | . . . 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 12525 | . . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑥) → 𝑥 = dom dom 𝐷) | |
14 | 13 | fveq2d 5425 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑥) → (∞Met‘𝑥) = (∞Met‘dom dom 𝐷)) |
15 | 12, 14 | eleqtrd 2218 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑥) → 𝐷 ∈ (∞Met‘dom dom 𝐷)) |
16 | 15 | rexlimivw 2545 | . . 3 ⊢ (∃𝑥 ∈ V 𝐷 ∈ (∞Met‘𝑥) → 𝐷 ∈ (∞Met‘dom dom 𝐷)) |
17 | 11, 16 | sylbi 120 | . 2 ⊢ (𝐷 ∈ ∪ ran ∞Met → 𝐷 ∈ (∞Met‘dom dom 𝐷)) |
18 | elex 2697 | . . . . . 6 ⊢ (𝐷 ∈ (∞Met‘dom dom 𝐷) → 𝐷 ∈ V) | |
19 | dmexg 4803 | . . . . . 6 ⊢ (𝐷 ∈ V → dom 𝐷 ∈ V) | |
20 | dmexg 4803 | . . . . . 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 5436 | . . . . 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 3096 | . . 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 125 | 1 ⊢ (𝐷 ∈ ∪ ran ∞Met ↔ 𝐷 ∈ (∞Met‘dom dom 𝐷)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ∀wral 2416 ∃wrex 2417 {crab 2420 Vcvv 2686 ⊆ wss 3071 ∪ cuni 3736 class class class wbr 3929 × cxp 4537 dom cdm 4539 ran crn 4540 Fn wfn 5118 ‘cfv 5123 (class class class)co 5774 ↑𝑚 cmap 6542 0cc0 7620 ℝ*cxr 7799 ≤ cle 7801 +𝑒 cxad 9557 ∞Metcxmet 12149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-map 6544 df-pnf 7802 df-mnf 7803 df-xr 7804 df-xmet 12157 |
This theorem is referenced by: isxms2 12621 |
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