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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 6633 | . . . . . . 7 ⊢ ↑𝑚 Fn (V × V) | |
2 | xrex 9813 | . . . . . . 7 ⊢ ℝ* ∈ V | |
3 | sqxpexg 4727 | . . . . . . . 8 ⊢ (𝑥 ∈ V → (𝑥 × 𝑥) ∈ V) | |
4 | 3 | elv 2734 | . . . . . . 7 ⊢ (𝑥 × 𝑥) ∈ V |
5 | fnovex 5886 | . . . . . . 7 ⊢ (( ↑𝑚 Fn (V × V) ∧ ℝ* ∈ V ∧ (𝑥 × 𝑥) ∈ V) → (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∈ V) | |
6 | 1, 2, 4, 5 | mp3an 1332 | . . . . . 6 ⊢ (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∈ V |
7 | 6 | rabex 4133 | . . . . 5 ⊢ {𝑑 ∈ (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))} ∈ V |
8 | df-xmet 12782 | . . . . 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 5326 | . . . 4 ⊢ ∞Met Fn V |
10 | fnunirn 5746 | . . . 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 13150 | . . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑥) → 𝑥 = dom dom 𝐷) | |
14 | 13 | fveq2d 5500 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑥) → (∞Met‘𝑥) = (∞Met‘dom dom 𝐷)) |
15 | 12, 14 | eleqtrd 2249 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑥) → 𝐷 ∈ (∞Met‘dom dom 𝐷)) |
16 | 15 | rexlimivw 2583 | . . 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 2741 | . . . . . 6 ⊢ (𝐷 ∈ (∞Met‘dom dom 𝐷) → 𝐷 ∈ V) | |
19 | dmexg 4875 | . . . . . 6 ⊢ (𝐷 ∈ V → dom 𝐷 ∈ V) | |
20 | dmexg 4875 | . . . . . 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 5511 | . . . . 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 3146 | . . 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 1348 ∈ wcel 2141 ∀wral 2448 ∃wrex 2449 {crab 2452 Vcvv 2730 ⊆ wss 3121 ∪ cuni 3796 class class class wbr 3989 × cxp 4609 dom cdm 4611 ran crn 4612 Fn wfn 5193 ‘cfv 5198 (class class class)co 5853 ↑𝑚 cmap 6626 0cc0 7774 ℝ*cxr 7953 ≤ cle 7955 +𝑒 cxad 9727 ∞Metcxmet 12774 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-map 6628 df-pnf 7956 df-mnf 7957 df-xr 7958 df-xmet 12782 |
This theorem is referenced by: isxms2 13246 |
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