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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 6612 | . . . . . . 7 ⊢ ↑𝑚 Fn (V × V) | |
2 | xrex 9783 | . . . . . . 7 ⊢ ℝ* ∈ V | |
3 | sqxpexg 4714 | . . . . . . . 8 ⊢ (𝑥 ∈ V → (𝑥 × 𝑥) ∈ V) | |
4 | 3 | elv 2725 | . . . . . . 7 ⊢ (𝑥 × 𝑥) ∈ V |
5 | fnovex 5866 | . . . . . . 7 ⊢ (( ↑𝑚 Fn (V × V) ∧ ℝ* ∈ V ∧ (𝑥 × 𝑥) ∈ V) → (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∈ V) | |
6 | 1, 2, 4, 5 | mp3an 1326 | . . . . . 6 ⊢ (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∈ V |
7 | 6 | rabex 4120 | . . . . 5 ⊢ {𝑑 ∈ (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))} ∈ V |
8 | df-xmet 12535 | . . . . 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 5310 | . . . 4 ⊢ ∞Met Fn V |
10 | fnunirn 5729 | . . . 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 12903 | . . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑥) → 𝑥 = dom dom 𝐷) | |
14 | 13 | fveq2d 5484 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑥) → (∞Met‘𝑥) = (∞Met‘dom dom 𝐷)) |
15 | 12, 14 | eleqtrd 2243 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑥) → 𝐷 ∈ (∞Met‘dom dom 𝐷)) |
16 | 15 | rexlimivw 2577 | . . 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 2732 | . . . . . 6 ⊢ (𝐷 ∈ (∞Met‘dom dom 𝐷) → 𝐷 ∈ V) | |
19 | dmexg 4862 | . . . . . 6 ⊢ (𝐷 ∈ V → dom 𝐷 ∈ V) | |
20 | dmexg 4862 | . . . . . 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 5495 | . . . . 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 3136 | . . 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 1342 ∈ wcel 2135 ∀wral 2442 ∃wrex 2443 {crab 2446 Vcvv 2721 ⊆ wss 3111 ∪ cuni 3783 class class class wbr 3976 × cxp 4596 dom cdm 4598 ran crn 4599 Fn wfn 5177 ‘cfv 5182 (class class class)co 5836 ↑𝑚 cmap 6605 0cc0 7744 ℝ*cxr 7923 ≤ cle 7925 +𝑒 cxad 9697 ∞Metcxmet 12527 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-map 6607 df-pnf 7926 df-mnf 7927 df-xr 7928 df-xmet 12535 |
This theorem is referenced by: isxms2 12999 |
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